Catalogue of GP/PARI Functions: Elliptic curves
Author: Karim Belabas
Keywords: PARI, GP, DOC

## Elliptic curves

#### Elliptic curve structures

An elliptic curve is given by a Weierstrass model

y^2+a_1xy+a_3y = x^3+a_2x^2+a_4x+a6,

whose discriminant is non-zero. Affine points on `E` are represented as two-component vectors `[x,y]`; the point at infinity, i.e. the identity element of the group law, is represented by the one-component vector `[0]`.

Given a vector of coefficients [a1,a2,a3,a4,a6], the function `ellinit` initializes and returns an ell structure. An additional optional argument allows to specify the base field in case it cannot be inferred from the curve coefficients. This structure contains data needed by elliptic curve related functions, and is generally passed as a first argument. Expensive data are skipped on initialization: they will be dynamically computed when (and if) needed, and then inserted in the structure. The precise layout of the ell structure is left undefined and should never be used directly. The following member functions are available, depending on the underlying domain.

#### Reduction

Let E be a curve defined over ℚp given by a p-integral model; if the curve has good reduction at p, we may define its reduction ~{E} over the finite field 𝔽p:

      ? E = ellinit([-3,1], O(5^10));  \\   E/ℚ5
      ? Et = ellinit(E, 5)
      ? ellcard(Et)  \\   ~{E}/𝔽5 has 7 points
      %3 = 7
      ? ellinit(E, 7)
       ***   at top-level: ellinit(E,7)
       ***                 ^ —  —  —  —
       *** ellinit: inconsistent moduli in ellinit: 5 != 7

Likewise, if a curve is defined over a number field K and 𝔭 is a maximal ideal with finite residue field 𝔽q, we define the reduction ~{E}/𝔽q provided E has good reduction at 𝔭. E/ℚ is an important special case:

      ? E = ellinit([-3,1]);
      ? factor(E.disc)
      %2 =
      [2 4]

      [3 4]
      ? Et = ellinit(E, 5);
      ? ellcard(Et) \\   ~{E} / 𝔽5 has 7 points
      %4 = 7
      ? ellinit(E, 3)  \\ bad reduction at 3
      %5 = []

General number fields are similar:

      ? K = nfinit(x^2+1); E = ellinit([x,x+1], K);
      ? idealfactor(K, E.disc)  \\ three primes of bad reduction
      %2 =
      [  [2, [1, 1]~, 2, 1, [1, -1; 1, 1]] 10]

      [ [5, [-2, 1]~, 1, 1, [2, -1; 1, 2]]  2]

      [[5, [2, 1]~, 1, 1, [-2, -1; 1, -2]]  2]
      ? P = idealprimedec(K, 3); \\ a prime of good reduction
      ? idealnorm(K, P)
      %4 = 9
      ? Et = ellinit(E, P);
      ? ellcard(Et)  \\   ~{E} / 𝔽9 has 4 points
      %6 = 4

If the model is not locally minimal at 𝔭, the above will fail: `elllocalred` and `ellchangecurve` allow to reduce to that case.

Some functions such as `ellap`, `ellcard`, `ellgroup` and `ellissupersingular` even implicitly replace the given equation by a local minimal model and consider the group of non-singular points ~{E}ns so they make sense even when the curve has bad reduction.

#### ellL1(E, {r = 0})

Returns the value at s = 1 of the derivative of order r of the L-function of the elliptic curve E.

      ? E = ellinit("11a1"); \\ order of vanishing is 0
      ? ellL1(E)
      %2 = 0.2538418608559106843377589233
      ? E = ellinit("389a1");  \\ order of vanishing is 2
      ? ellL1(E)
      %4 = -5.384067311837218089235032414 E-29
      ? ellL1(E, 1)
      %5 = 0
      ? ellL1(E, 2)
      %6 = 1.518633000576853540460385214

The main use of this function, after computing at low accuracy the order of vanishing using `ellanalyticrank`, is to compute the leading term at high accuracy to check (or use) the Birch and Swinnerton-Dyer conjecture:

      ? \p18
        realprecision = 18 significant digits
      ? E = ellinit("5077a1"); ellanalyticrank(E)
      time = 8 ms.
      %1 = [3, 10.3910994007158041]
      ? \p200
        realprecision = 202 significant digits (200 digits displayed)
      ? ellL1(E, 3)
      time = 104 ms.
      %3 = 10.3910994007158041387518505103609170697263563756570092797 [...]

The library syntax is `GEN ellL1_bitprec(GEN E, long r, long bitprec)`.

#### elladd(E, z1, z2)

Sum of the points z1 and z2 on the elliptic curve corresponding to E.

The library syntax is `GEN elladd(GEN E, GEN z1, GEN z2)`.

#### ellak(E, n)

Computes the coefficient an of the L-function of the elliptic curve E/ℚ, i.e. coefficients of a newform of weight 2 by the modularity theorem (Taniyama-Shimura-Weil conjecture). E must be an `ell` structure over ℚ as output by `ellinit`. E must be given by an integral model, not necessarily minimal, although a minimal model will make the function faster.

      ? E = ellinit([1,-1,0,4,3]);
      ? ellak(E, 10)
      %2 = -3
      ? e = ellchangecurve(E, [1/5,0,0,0]); \\ made not minimal at 5
      ? ellak(e, 10) \\ wasteful but works
      %3 = -3
      ? E = ellminimalmodel(e); \\ now minimal
      ? ellak(E, 5)
      %5 = -3

If the model is not minimal at a number of bad primes, then the function will be slower on those n divisible by the bad primes. The speed should be comparable for other n:

      ? for(i=1,10^6, ellak(E,5))
      time = 699 ms.
      ? for(i=1,10^6, ellak(e,5)) \\ 5 is bad, markedly slower
      time = 1,079 ms.

      ? for(i=1,10^5,ellak(E,5*i))
      time = 1,477 ms.
      ? for(i=1,10^5,ellak(e,5*i)) \\ still slower but not so much on average
      time = 1,569 ms.

The library syntax is `GEN akell(GEN E, GEN n)`.

#### ellan(E, n)

Computes the vector of the first n Fourier coefficients ak corresponding to the elliptic curve E defined over a number field. If E is defined over ℚ, the curve may be given by an arbitrary model, not necessarily minimal, although a minimal model will make the function faster. Over a more general number field, the model must be locally minimal at all primes above 2 and 3.

The library syntax is `GEN ellan(GEN E, long n)`. Also available is `GEN ellanQ_zv(GEN e, long n)`, which returns a `t_VECSMALL` instead of a `t_VEC`, saving on memory.

#### ellanalyticrank(E, {eps})

Returns the order of vanishing at s = 1 of the L-function of the elliptic curve E and the value of the first non-zero derivative. To determine this order, it is assumed that any value less than `eps` is zero. If `eps` is omitted, 2-b/2 is used, where b is the current bit precision.

      ? E = ellinit("11a1"); \\ rank 0
      ? ellanalyticrank(E)
      %2 = [0, 0.2538418608559106843377589233]
      ? E = ellinit("37a1"); \\ rank 1
      ? ellanalyticrank(E)
      %4 = [1, 0.3059997738340523018204836835]
      ? E = ellinit("389a1"); \\ rank 2
      ? ellanalyticrank(E)
      %6 = [2, 1.518633000576853540460385214]
      ? E = ellinit("5077a1"); \\ rank 3
      ? ellanalyticrank(E)
      %8 = [3, 10.39109940071580413875185035]

The library syntax is `GEN ellanalyticrank_bitprec(GEN E, GEN eps = NULL, long bitprec)`.

#### ellap(E, {p})

Let `E` be an `ell` structure as output by `ellinit`, attached to an elliptic curve E/K. If the field K = 𝔽q is finite, return the trace of Frobenius t, defined by the equation #E(𝔽q) = q+1 - t.

For other fields of definition and p defining a finite residue field 𝔽q, return the trace of Frobenius for the reduction of E: the argument p is best left omitted if K = ℚ_ℓ (else we must have p = ℓ) and must be a prime number (K = ℚ) or prime ideal (K a general number field) with residue field 𝔽q otherwise. The equation need not be minimal or even integral at p; of course, a minimal model will be more efficient.

For a number field K, the trace of Frobenius is the ap coefficient in the Euler product defining the curve L-series, whence the function name: L(E/K,s) = ∏bad p (1-ap (Np)-s)-1 ∏good p (1-ap (Np)-s + (Np)1-2s)-1.

When the characteristic of the finite field is large, the availability of the `seadata` package will speed up the computation.

      ? E = ellinit([0,1]);  \\ y^2 = x^3 + 0.x + 1, defined over Q
      ? ellap(E, 7) \\ 7 necessary here
      %2 = -4       \\ #E(F7) = 7+1-(-4) = 12
      ? ellcard(E, 7)
      %3 = 12       \\ OK

      ? E = ellinit([0,1], 11);  \\ defined over F_11
      ? ellap(E)       \\ no need to repeat 11
      %4 = 0
      ? ellap(E, 11)   \\ ... but it also works
      %5 = 0
      ? ellgroup(E, 13) \\ ouch, inconsistent input!
         ***   at top-level: ellap(E,13)
         ***                 ^ —  —  — --
         *** ellap: inconsistent moduli in Rg_to_Fp:
           11
           13
      ? a = ffgen(ffinit(11,3), 'a); \\ defines Fq := F11^3
      ? E = ellinit([a+1,a]);  \\ y^2 = x^3 + (a+1)x + a, defined over Fq
      ? ellap(E)
      %8 = -3

If the curve is defined over a more general number field than ℚ, the maximal ideal p must be explicitly given in `idealprimedec` format. There is no assumption of local minimality at p.

      ? K = nfinit(a^2+1); E = ellinit([1+a,0,1,0,0], K);
      ? fa = idealfactor(K, E.disc)
      %2 =
      [  [5, [-2, 1]~, 1, 1, [2, -1; 1, 2]] 1]

      [[13, [5, 1]~, 1, 1, [-5, -1; 1, -5]] 2]
      ? ellap(E, fa[1,1])
      %3 = -1 \\ non-split multiplicative reduction
      ? ellap(E, fa[2,1])
      %4 = 1  \\ split multiplicative reduction
      ? P17 = idealprimedec(K,17)[1];
      ? ellap(E, P17)
      %6 = 6  \\ good reduction
      ? E2 = ellchangecurve(E, [17,0,0,0]);
      ? ellap(E2, P17)
      %8 = 6  \\ same, starting from a non-miminal model

      ? P3 = idealprimedec(K,3)[1];
      ? ellap(E, P3)  \\ OK: E is minimal at P3
      %10 = -2
      ? E3 = ellchangecurve(E, [3,0,0,0]);
      ? ellap(E3, P3) \\ not integral at P3
       ***   at top-level: ellap(E3,P3)
       ***                 ^ —  —  —  —
       *** ellap: impossible inverse in Rg_to_ff: Mod(0, 3).

Algorithms used. If E/𝔽q has CM by a principal imaginary quadratic order we use a fast explicit formula (involving essentially Kronecker symbols and Cornacchia's algorithm), in O(log q)^2 bit operations. Otherwise, we use Shanks-Mestre's baby-step/giant-step method, which runs in time Õ(q1/4) using Õ(q1/4) storage, hence becomes unreasonable when q has about 30 digits. Above this range, the `SEA` algorithm becomes available, heuristically in Õ(log q)^4, and primes of the order of 200 digits become feasible. In small characteristic we use Mestre's (p = 2), Kohel's (p = 3,5,7,13), Satoh-Harley (all in Õ(p2 n^2)) or Kedlaya's (in Õ(p n^3)) algorithms.

The library syntax is `GEN ellap(GEN E, GEN p = NULL)`.

#### ellbil(E, z1, z2)

Deprecated alias for `ellheight(E,P,Q)`.

The library syntax is `GEN bilhell(GEN E, GEN z1, GEN z2, long prec)`.

#### ellbsd(E)

The object E being an elliptic curve over a number field, returns a real number c such that the BSD conjecture predicts that LE(r)(1)/r != c R S where r is the rank, R the regulator and S the cardinal of the Tate-Shafarevich group.

      ? e = ellinit([0,-1,1,-10,-20]); \\ rank 0
      ? ellbsd(e)
      %2 = 0.25384186085591068433775892335090946105
      ? lfun(e,1)
      %3 = 0.25384186085591068433775892335090946104
      ? e = ellinit([0,0,1,-1,0]); \\ rank 1
      ? P = ellheegner(e);
      ? ellbsd(e)*ellheight(e,P)
      %6 = 0.30599977383405230182048368332167647445
      ? lfun(e,1,1)
      %7 = 0.30599977383405230182048368332167647445
      ? e = ellinit([1+a,0,1,0,0],nfinit(a^2+1)); \\ rank 0
      ? ellbsd(e)
      %9 = 0.42521832235345764503001271536611593310
      ? lfun(e,1)
      %10 = 0.42521832235345764503001271536611593309

The library syntax is `GEN ellbsd(GEN E, long prec)`.

#### ellcard(E, {p})

Let `E` be an `ell` structure as output by `ellinit`, attached to an elliptic curve E/K. If K = 𝔽q is finite, return the order of the group E(𝔽q).

      ? E = ellinit([-3,1], 5); ellcard(E)
      %1 = 7
      ? t = ffgen(3^5,'t); E = ellinit([t,t^2+1]); ellcard(E)
      %2 = 217

For other fields of definition and p defining a finite residue field 𝔽q, return the order of the reduction of E: the argument p is best left omitted if K = ℚ_ℓ (else we must have p = ℓ) and must be a prime number (K = ℚ) or prime ideal (K a general number field) with residue field 𝔽q otherwise. The equation need not be minimal or even integral at p; of course, a minimal model will be more efficient. The function considers the group of non-singular points of the reduction of a minimal model of the curve at p, so also makes sense when the curve has bad reduction.

      ? E = ellinit([-3,1]);
      ? factor(E.disc)
      %2 =
      [2 4]

      [3 4]
      ? ellcard(E, 5)  \\ as above !
      %3 = 7
      ? ellcard(E, 2) \\ additive reduction
      %4 = 2

When the characteristic of the finite field is large, the availability of the `seadata` package will speed the computation. See also `ellap` for the list of implemented algorithms.

The library syntax is `GEN ellcard(GEN E, GEN p = NULL)`. Also available is `GEN ellcard(GEN E, GEN p)` where p is not `NULL`.

#### ellchangecurve(E, v)

Changes the data for the elliptic curve E by changing the coordinates using the vector `v = [u,r,s,t]`, i.e. if x' and y' are the new coordinates, then x = u^2x'+r, y = u^3y'+su^2x'+t. E must be an `ell` structure as output by `ellinit`. The special case v = 1 is also used instead of [1,0,0,0] to denote the trivial coordinate change.

The library syntax is `GEN ellchangecurve(GEN E, GEN v)`.

#### ellchangepoint(x, v)

Changes the coordinates of the point or vector of points x using the vector `v = [u,r,s,t]`, i.e. if x' and y' are the new coordinates, then x = u^2x'+r, y = u^3y'+su^2x'+t (see also `ellchangecurve`).

      ? E0 = ellinit([1,1]); P0 = [0,1]; v = [1,2,3,4];
      ? E = ellchangecurve(E0, v);
      ? P = ellchangepoint(P0,v)
      %3 = [-2, 3]
      ? ellisoncurve(E, P)
      %4 = 1
      ? ellchangepointinv(P,v)
      %5 = [0, 1]

The library syntax is `GEN ellchangepoint(GEN x, GEN v)`. The reciprocal function `GEN ellchangepointinv(GEN x, GEN ch)` inverts the coordinate change.

#### ellchangepointinv(x, v)

Changes the coordinates of the point or vector of points x using the inverse of the isomorphism attached to `v = [u,r,s,t]`, i.e. if x' and y' are the old coordinates, then x = u^2x'+r, y = u^3y'+su^2x'+t (inverse of `ellchangepoint`).

      ? E0 = ellinit([1,1]); P0 = [0,1]; v = [1,2,3,4];
      ? E = ellchangecurve(E0, v);
      ? P = ellchangepoint(P0,v)
      %3 = [-2, 3]
      ? ellisoncurve(E, P)
      %4 = 1
      ? ellchangepointinv(P,v)
      %5 = [0, 1]  \\ we get back P0

The library syntax is `GEN ellchangepointinv(GEN x, GEN v)`.

#### ellconvertname(name)

Converts an elliptic curve name, as found in the `elldata` database, from a string to a triplet [conductor, isogeny class, index]. It will also convert a triplet back to a curve name. Examples:

      ? ellconvertname("123b1")
      %1 = [123, 1, 1]
      ? ellconvertname(%)
      %2 = "123b1"

The library syntax is `GEN ellconvertname(GEN name)`.

#### elldivpol(E, n, {v = 'x})

n-division polynomial fn for the curve E in the variable v. In standard notation, for any affine point P = (X,Y) on the curve, we have [n]P = (φn(P)ψn(P) : ωn(P) : ψn(P)^3) for some polynomials φn,ωn,ψn in ℤ[a1,a2,a3,a4,a6][X,Y]. We have fn(X) = ψn(X) for n odd, and fn(X) = ψn(X,Y) (2Y + a_1X+a3) for n even. We have f1 = 1, f2 = 4X^3 + b_2X^2 + 2b4 X + b6, f3 = 3 X^4 + b2 X^3 + 3b4 X^2 + 3 b6 X + b8, f4 = f2(2X^6 + b2 X^5 + 5b4 X^4 + 10 b6 X^3 + 10 b8 X^2 + (b_2b8-b_4b6)X + (b_8b4 - b6^2)),... For n ≥ 2, the roots of fn are the X-coordinates of points in E[n].

The library syntax is `GEN elldivpol(GEN E, long n, long v = -1)` where `v` is a variable number.

#### elleisnum(w, k, {flag = 0})

k being an even positive integer, computes the numerical value of the Eisenstein series of weight k at the lattice w, as given by `ellperiods`, namely

(2i π/ω2)^k (1 + 2/ζ(1-k) ∑n ≥ 1 nk-1q^n / (1-q^n)),

where q = exp(2iπ τ) and τ := ω1/ω2 belongs to the complex upper half-plane. It is also possible to directly input w = [ω1,ω2], or an elliptic curve E as given by `ellinit`.

      ? w = ellperiods([1,I]);
      ? elleisnum(w, 4)
      %2 = 2268.8726415508062275167367584190557607
      ? elleisnum(w, 6)
      %3 = -3.977978632282564763 E-33
      ? E = ellinit([1, 0]);
      ? elleisnum(E, 4, 1)
      %5 = -47.999999999999999999999999999999999998

When flag is non-zero and k = 4 or 6, returns the elliptic invariants g2 or g3, such that y^2 = 4x^3 - g2 x - g3 is a Weierstrass equation for E.

The library syntax is `GEN elleisnum(GEN w, long k, long flag, long prec)`.

#### elleta(w)

Returns the quasi-periods [η1,η2] attached to the lattice basis w = [ω1, ω2]. Alternatively, w can be an elliptic curve E as output by `ellinit`, in which case, the quasi periods attached to the period lattice basis `E.omega` (namely, `E.eta`) are returned.

      ? elleta([1, I])
      %1 = [3.141592653589793238462643383, 9.424777960769379715387930149*I]

The library syntax is `GEN elleta(GEN w, long prec)`.

#### ellformaldifferential(E, {n = seriesprecision}, {t = 'x})

Let ω := dx / (2y+a_1x+a3) be the invariant differential form attached to the model E of some elliptic curve (`ellinit` form), and η := x(t)ω. Return n terms (`seriesprecision` by default) of f(t),g(t) two power series in the formal parameter t = -x/y such that ω = f(t) dt, η = g(t) dt: f(t) = 1+a1 t + (a1^2 + a2) t^2 +..., g(t) = t-2 +...

       ? E = ellinit([-1,1/4]); [f,g] = ellformaldifferential(E,7,'t);
       ? f
       %2 = 1 - 2*t^4 + 3/4*t^6 + O(t^7)
       ? g
       %3 = t^-2 - t^2 + 1/2*t^4 + O(t^5)

The library syntax is `GEN ellformaldifferential(GEN E, long precdl, long n = -1)` where `n` is a variable number.

#### ellformalexp(E, {n = seriesprecision}, {z = 'x})

The elliptic formal exponential `Exp` attached to E is the isomorphism from the formal additive law to the formal group of E. It is normalized so as to be the inverse of the elliptic logarithm (see `ellformallog`): `Exp` o L = Id. Return n terms of this power series:

      ? E=ellinit([-1,1/4]); Exp = ellformalexp(E,10,'z)
      %1 = z + 2/5*z^5 - 3/28*z^7 + 2/15*z^9 + O(z^11)
      ? L = ellformallog(E,10,'t);
      ? subst(Exp,z,L)
      %3 = t + O(t^11)

The library syntax is `GEN ellformalexp(GEN E, long precdl, long n = -1)` where `n` is a variable number.

#### ellformallog(E, {n = seriesprecision}, {v = 'x})

The formal elliptic logarithm is a series L in t K[[t]] such that d L = ω = dx / (2y + a_1x + a3), the canonical invariant differential attached to the model E. It gives an isomorphism from the formal group of E to the additive formal group.

      ? E = ellinit([-1,1/4]); L = ellformallog(E, 9, 't)
      %1 = t - 2/5*t^5 + 3/28*t^7 + 2/3*t^9 + O(t^10)
      ? [f,g] = ellformaldifferential(E,8,'t);
      ? L' - f
      %3 = O(t^8)

The library syntax is `GEN ellformallog(GEN E, long precdl, long n = -1)` where `n` is a variable number.

#### ellformalpoint(E, {n = seriesprecision}, {v = 'x})

If E is an elliptic curve, return the coordinates x(t), y(t) in the formal group of the elliptic curve E in the formal parameter t = -x/y at oo : x = t-2 -a1 t-1 - a2 - a3 t +... y = - t-3 -a1 t-2 - a_2t-1 -a3 +... Return n terms (`seriesprecision` by default) of these two power series, whose coefficients are in ℤ[a1,a2,a3,a4,a6].

      ? E = ellinit([0,0,1,-1,0]); [x,y] = ellformalpoint(E,8,'t);
      ? x
      %2 = t^-2 - t + t^2 - t^4 + 2*t^5 + O(t^6)
      ? y
      %3 = -t^-3 + 1 - t + t^3 - 2*t^4 + O(t^5)
      ? E = ellinit([0,1/2]); ellformalpoint(E,7)
      %4 = [x^-2 - 1/2*x^4 + O(x^5), -x^-3 + 1/2*x^3 + O(x^4)]

The library syntax is `GEN ellformalpoint(GEN E, long precdl, long n = -1)` where `n` is a variable number.

#### ellformalw(E, {n = seriesprecision}, {t = 'x})

Return the formal power series w attached to the elliptic curve E, in the variable t: w(t) = t^3 + a1 t^4 + (a2 + a1^2) t^5 +...+ O(tn+3), which is the formal expansion of -1/y in the formal parameter t := -x/y at oo (take n = `seriesprecision` if n is omitted). The coefficients of w belong to ℤ[a1,a2,a3,a4,a6].

      ? E=ellinit([3,2,-4,-2,5]); ellformalw(E, 5, 't)
      %1 = t^3 + 3*t^4 + 11*t^5 + 35*t^6 + 101*t^7 + O(t^8)

The library syntax is `GEN ellformalw(GEN E, long precdl, long n = -1)` where `n` is a variable number.

#### ellfromeqn(P)

Given a genus 1 plane curve, defined by the affine equation f(x,y) = 0, return the coefficients [a1,a2,a3,a4,a6] of a Weierstrass equation for its Jacobian. This allows to recover a Weierstrass model for an elliptic curve given by a general plane cubic or by a binary quartic or biquadratic model. The function implements the f `⟼` f* formulae of Artin, Tate and Villegas (Advances in Math. 198 (2005), pp. 366--382).

In the example below, the function is used to convert between twisted Edwards coordinates and Weierstrass coordinates.

      ? e = ellfromeqn(a*x^2+y^2 - (1+d*x^2*y^2))
      %1 = [0, -a - d, 0, -4*d*a, 4*d*a^2 + 4*d^2*a]
      ? E = ellinit(ellfromeqn(y^2-x^2 - 1 +(121665/121666*x^2*y^2)),2^255-19);
      ? isprime(ellcard(E) / 8)
      %3 = 1

The elliptic curve attached to the sum of two cubes is given by

      ? ellfromeqn(x^3+y^3 - a)
      %1 = [0, 0, -9*a, 0, -27*a^2]

Congruent number problem:. Let n be an integer, if a^2+b^2 = c^2 and a b = 2 n, then by substituting b by 2 n/a in the first equation, we get ((a^2+(2 n/a)^2)-c^2) a^2 = 0. We set x = a, y = a c.

      ? En = ellfromeqn((x^2 + (2*n/x)^2 - (y/x)^2)*x^2)
      %1 = [0, 0, 0, -16*n^2, 0]

For example 23 is congruent since the curve has a point of infinite order, namely:

      ? ellheegner( ellinit(subst(En, n, 23)) )
      %2 = [168100/289, 68053440/4913]

The library syntax is `GEN ellfromeqn(GEN P)`.

#### ellfromj(j)

Returns the coefficients [a1,a2,a3,a4,a6] of a fixed elliptic curve with j-invariant j.

The library syntax is `GEN ellfromj(GEN j)`.

#### ellgenerators(E)

If E is an elliptic curve over the rationals, return a ℤ-basis of the free part of the Mordell-Weil group attached to E. This relies on the `elldata` database being installed and referencing the curve, and so is only available for curves over ℤ of small conductors. If E is an elliptic curve over a finite field 𝔽q as output by `ellinit`, return a minimal set of generators for the group E(𝔽q).

Caution. when the group is not cyclic, of shape ℤ/d1ℤ x ℤ/d2ℤ with d2 | d1, the points [P,Q] returned by ellgenerators need not have order d1 and d2: it is true that P has order d1, but we only know that Q is a generator of E(𝔽q)/ <P> and that the Weil pairing w(P,Q) has order d2, see `??ellgroup`. If you need generators [P,R] with R of order d2, find x such that R = Q-[x]P has order d2 by solving the discrete logarithm problem [d2]Q = [x]([d2]P) in a cyclic group of order d1/d2. This will be very expensive if d1/d2 has a large prime factor.

The library syntax is `GEN ellgenerators(GEN E)`.

#### ellglobalred(E)

Let E be an `ell` structure as output by `ellinit` attached to an elliptic curve defined over a number field. This function calculates the arithmetic conductor and the global Tamagawa number c. The result [N,v,c,F,L] is slightly different if E is defined over ℚ (domain D = 1 in `ellinit`) or over a number field (domain D is a number field structure, including `nfinit(x)` representing ℚ !):

* N is the arithmetic conductor of the curve,

* v is an obsolete field, left in place for backward compatibility. If E is defined over ℚ, v gives the coordinate change for E to the standard minimal integral model (`ellminimalmodel` provides it in a cheaper way); if E is defined over another number field, v gives a coordinate change to an integral model (`ellintegralmodel` provides it in a cheaper way).

* c is the product of the local Tamagawa numbers cp, a quantity which enters in the Birch and Swinnerton-Dyer conjecture,

* F is the factorization of N,

* L is a vector, whose i-th entry contains the local data at the i-th prime ideal divisor of N, i.e. `L[i] = elllocalred(E,F[i,1])`. If E is defined over ℚ, the local coordinate change has been deleted and replaced by a 0; if E is defined over another number field the local coordinate change to a local minimal model is given relative to the integral model afforded by v (so either start from an integral model so that v be trivial, or apply v first).

The library syntax is `GEN ellglobalred(GEN E)`.

#### ellgroup(E, {p}, {flag})

Let `E` be an `ell` structure as output by `ellinit`, attached to an elliptic curve E/K. We first describle the function when the field K = 𝔽q is finite, it computes the structure of the finite abelian group E(𝔽q):

* if flag = 0, return the structure [] (trivial group) or [d1] (non-trivial cyclic group) or [d1,d2] (non-cyclic group) of E(𝔽q) ~ ℤ/d1ℤ x ℤ/d2ℤ, with d2 | d1.

* if flag = 1, return a triple [h,cyc,gen], where h is the curve cardinality, cyc gives the group structure as a product of cyclic groups (as per flag = 0). More precisely, if d2 > 1, the output is [d_1d2, [d1,d2], [P,Q]] where P is of order d1 and [P,Q] generates the curve. Caution. It is not guaranteed that Q has order d2, which in the worst case requires an expensive discrete log computation. Only that `ellweilpairing`(E, P, Q, d1) has order d2.

For other fields of definition and p defining a finite residue field 𝔽q, return the structure of the reduction of E: the argument p is best left omitted if K = ℚ_ℓ (else we must have p = ℓ) and must be a prime number (K = ℚ) or prime ideal (K a general number field) with residue field 𝔽q otherwise. The curve is allowed to have bad reduction at p and in this case we consider the (cyclic) group of non-singular points for the reduction of a minimal model at p.

If flag = 0, the equation not be minimal or even integral at p; of course, a minimal model will be more efficient.

If flag = 1, the requested generators depend on the model, which must then be minimal at p, otherwise an exception is thrown. Use `ellintegralmodel` and/or `ellocalred` first to reduce to this case.

      ? E = ellinit([0,1]);  \\ y^2 = x^3 + 0.x + 1, defined over Q
      ? ellgroup(E, 7)
      %2 = [6, 2] \\ Z/6 x Z/2, non-cyclic
      ? E = ellinit([0,1] * Mod(1,11));  \\ defined over F_11
      ? ellgroup(E)   \\ no need to repeat 11
      %4 = [12]
      ? ellgroup(E, 11)   \\ ... but it also works
      %5 = [12]
      ? ellgroup(E, 13) \\ ouch, inconsistent input!
         ***   at top-level: ellgroup(E,13)
         ***                 ^ —  —  —  — --
         *** ellgroup: inconsistent moduli in Rg_to_Fp:
           11
           13
      ? ellgroup(E, 7, 1)
      %6 = [12, [6, 2], [[Mod(2, 7), Mod(4, 7)], [Mod(4, 7), Mod(4, 7)]]]

Let us now consider curves of bad reduction, in this case we return the structure of the (cyclic) group of non-singular points, satisfying #Ens(𝔽p) = p - ap:

      ? E = ellinit([0,5]);
      ? ellgroup(E, 5, 1)
      %2 = [5, [5], [[Mod(4, 5), Mod(2, 5)]]]
      ? ellap(E, 5)
      %3 = 0 \\ additive reduction at 5
      ? E = ellinit([0,-1,0,35,0]);
      ? ellgroup(E, 5, 1)
      %5 = [4, [4], [[Mod(2, 5), Mod(2, 5)]]]
      ? ellap(E, 5)
      %6 = 1 \\ split multiplicative reduction at 5
      ? ellgroup(E, 7, 1)
      %7 = [8, [8], [[Mod(3, 7), Mod(5, 7)]]]
      ? ellap(E, 7)
      %8 = -1 \\ non-split multiplicative reduction at 7

The library syntax is `GEN ellgroup0(GEN E, GEN p = NULL, long flag)`. Also available is `GEN ellgroup(GEN E, GEN p)`, corresponding to flag = 0.

#### ellheegner(E)

Let E be an elliptic curve over the rationals, assumed to be of (analytic) rank 1. This returns a non-torsion rational point on the curve, whose canonical height is equal to the product of the elliptic regulator by the analytic Sha.

This uses the Heegner point method, described in Cohen GTM 239; the complexity is proportional to the product of the square root of the conductor and the height of the point (thus, it is preferable to apply it to strong Weil curves).

      ? E = ellinit([-157^2,0]);
      ? u = ellheegner(E); print(u[1], "\n", u[2])
      69648970982596494254458225/166136231668185267540804
      538962435089604615078004307258785218335/67716816556077455999228495435742408
      ? ellheegner(ellinit([0,1]))         \\ E has rank 0 !
       ***   at top-level: ellheegner(E=ellinit
       ***                 ^ —  —  —  —  —  — --
       *** ellheegner: The curve has even analytic rank.

The library syntax is `GEN ellheegner(GEN E)`.

#### ellheight(E, P, {Q})

Global Néron-Tate height h(P) of the point P on the elliptic curve E, defined over ℚ or a number field, using the normalization in Cremona's Algorithms for modular elliptic curves. E must be an `ell` as output by `ellinit`; it needs not be given by a minimal model although the computation will be faster if it is.

If the argument Q is present, computes the value of the bilinear form (h(P+Q)-h(P-Q)) / 4.

The library syntax is `GEN ellheight0(GEN E, GEN P, GEN Q = NULL, long prec)`. Also available is `GEN ellheight(GEN E, GEN P, long prec)` (Q omitted).

#### ellheightmatrix(E, x)

x being a vector of points, this function outputs the Gram matrix of x with respect to the Néron-Tate height, in other words, the (i,j) component of the matrix is equal to `ellbil(E,x[i],x[j])`. The rank of this matrix, at least in some approximate sense, gives the rank of the set of points, and if x is a basis of the Mordell-Weil group of E, its determinant is equal to the regulator of E. Note our height normalization follows Cremona's Algorithms for modular elliptic curves: this matrix should be divided by 2 to be in accordance with, e.g., Silverman's normalizations.

The library syntax is `GEN ellheightmatrix(GEN E, GEN x, long prec)`.

#### ellidentify(E)

Look up the elliptic curve E, defined by an arbitrary model over ℚ, in the `elldata` database. Return `[[N, M, G], C]` where N is the curve name in Cremona's elliptic curve database, M is the minimal model, G is a ℤ-basis of the free part of the Mordell-Weil group E(ℚ) and C is the change of coordinates change, suitable for `ellchangecurve`.

The library syntax is `GEN ellidentify(GEN E)`.

#### ellinit(x, {D = 1})

Initialize an `ell` structure, attached to the elliptic curve E. E is either

* a 5-component vector [a1,a2,a3,a4,a6] defining the elliptic curve with Weierstrass equation Y^2 + a1 XY + a3 Y = X^3 + a2 X^2 + a4 X + a6,

* a 2-component vector [a4,a6] defining the elliptic curve with short Weierstrass equation Y^2 = X^3 + a4 X + a6,

* a character string in Cremona's notation, e.g. `"11a1"`, in which case the curve is retrieved from the `elldata` database if available.

The optional argument D describes the domain over which the curve is defined:

* the `t_INT` 1 (default): the field of rational numbers ℚ.

* a `t_INT` p, where p is a prime number: the prime finite field 𝔽p.

* an `t_INTMOD` `Mod(a, p)`, where p is a prime number: the prime finite field 𝔽p.

* a `t_FFELT`, as returned by `ffgen`: the corresponding finite field 𝔽q.

* a `t_PADIC`, O(p^n): the field ℚp, where p-adic quantities will be computed to a relative accuracy of n digits. We advise to input a model defined over ℚ for such curves. In any case, if you input an approximate model with `t_PADIC` coefficients, it will be replaced by a lift to ℚ (an exact model "close" to the one that was input) and all quantities will then be computed in terms of this lifted model, at the given accuracy.

* a `t_REAL` x: the field ℂ of complex numbers, where floating point quantities are by default computed to a relative accuracy of `precision`(x). If no such argument is given, the value of `realprecision` at the time `ellinit` is called will be used.

* a number field K, given by a `nf` or `bnf` structure; a `bnf` is required for `ellminimalmodel`.

* a prime ideal 𝔭, given by a `prid` structure; valid if x is a curve defined over a number field K and the equation is integral and minimal at 𝔭.

This argument D is indicative: the curve coefficients are checked for compatibility, possibly changing D; for instance if D = 1 and an `t_INTMOD` is found. If inconsistencies are detected, an error is raised:

      ? ellinit([1 + O(5), 1], O(7));
       ***   at top-level: ellinit([1+O(5),1],O
       ***                 ^ —  —  —  —  —  — --
       *** ellinit: inconsistent moduli in ellinit: 7 != 5

If the curve coefficients are too general to fit any of the above domain categories, only basic operations, such as point addition, will be supported later.

If the curve (seen over the domain D) is singular, fail and return an empty vector [].

      ? E = ellinit([0,0,0,0,1]); \\ y^2 = x^3 + 1, over Q
      ? E = ellinit([0,1]);       \\ the same curve, short form
      ? E = ellinit("36a1");      \\ sill the same curve, Cremona's notations
      ? E = ellinit([0,1], 2)     \\ over F2: singular curve
      %4 = []
      ? E = ellinit(['a4,'a6] * Mod(1,5));  \\ over F5[a4,a6], basic support !

The result of `ellinit` is an ell structure. It contains at least the following information in its components:

a1,a2,a3,a4,a6,b2,b4,b6,b8,c4,c6,Δ,j.

All are accessible via member functions. In particular, the discriminant is `E.disc`, and the j-invariant is `E.j`.

      ? E = ellinit([a4, a6]);
      ? E.disc
      %2 = -64*a4^3 - 432*a6^2
      ? E.j
      %3 = -6912*a4^3/(-4*a4^3 - 27*a6^2)

Further components contain domain-specific data, which are in general dynamic: only computed when needed, and then cached in the structure.

      ? E = ellinit([2,3], 10^60+7);  \\ E over Fp, p large
      ? ellap(E)
      time = 4,440 ms.
      %2 = -1376268269510579884904540406082
      ? ellcard(E);  \\ now instantaneous !
      time = 0 ms.
      ? ellgenerators(E);
      time = 5,965 ms.
      ? ellgenerators(E); \\ second time instantaneous
      time = 0 ms.

See the description of member functions related to elliptic curves at the beginning of this section.

The library syntax is `GEN ellinit(GEN x, GEN D = NULL, long prec)`.

#### ellintegralmodel(E, {&v})

Let E be an `ell` structure over a number field K or ℚp. This function returns an integral model. If v is present, sets v = [u,0,0,0] to the corresponding change of variable: the return value is identical to that of `ellchangecurve(E, v)`.

      ? e = ellinit([1/17,1/42]);
      ? e = ellintegralmodel(e,&v);
      ? e[1..5]
      %3 = [0, 0, 0, 15287762448, 3154568630095008]
      ? v
      %4 = [1/714, 0, 0, 0]

The library syntax is `GEN ellintegralmodel(GEN E, GEN *v = NULL)`.

#### ellisdivisible(E, P, n, {&Q})

Given E/K a number field and P in E(K) return 1 if P = [n]R for some R in E(K) and set Q to one such R; and return 0 otherwise. The integer n ≥ 0 may be given as `ellxn(E,n)`, if many points need to be tested.

      ? K = nfinit(polcyclo(11,t));
      ? E = ellinit([0,-1,1,0,0], K);
      ? P = [0,0];
      ? ellorder(E,P)
      %4 = 5
      ? ellisdivisible(E,P,5, &Q)
      %5 = 1
      ? lift(Q)
      %6 = [-t^7-t^6-t^5-t^4+1, -t^9-2*t^8-2*t^7-3*t^6-3*t^5-2*t^4-2*t^3-t^2-1]
      ? ellorder(E, Q)
      %7 = 25

The algebraic complexity of the underlying algorithm is in O(n^4), so it is advisable to first factor n, then use a chain of checks attached to the prime divisors of n: the function will do it itself unless n is given in `ellxn` form.

The library syntax is `long ellisdivisible(GEN E, GEN P, GEN n, GEN *Q = NULL)`.

#### ellisogeny(E, G, {only_image = 0}, {x = 'x}, {y = 'y})

Given an elliptic curve E, a finite subgroup G of E is given either as a generating point P (for a cyclic G) or as a polynomial whose roots vanish on the x-coordinates of the non-zero elements of G (general case and more efficient if available). This function returns the [a1,a2,a3,a4,a6] invariants of the quotient elliptic curve E/G and (if only_image is zero (the default)) a vector of rational functions [f, g, h] such that the isogeny E → E/G is given by (x,y) `: — >`(f(x)/h(x)^2, g(x,y)/h(x)^3).

      ? E = ellinit([0,1]);
      ? elltors(E)
      %2 = [6, [6], [[2, 3]]]
      ? ellisogeny(E, [2,3], 1)  \\ Weierstrass model for E/<P>
      %3 = [0, 0, 0, -135, -594]
      ? ellisogeny(E,[-1,0])
      %4 = [[0,0,0,-15,22], [x^3+2*x^2+4*x+3, y*x^3+3*y*x^2-2*y, x+1]]

The library syntax is `GEN ellisogeny(GEN E, GEN G, long only_image, long x = -1, long y = -1)` where `x`, `y` are variable numbers.

#### ellisogenyapply(f, g)

Given an isogeny of elliptic curves f:E' → E (being the result of a call to `ellisogeny`), apply f to g:

* if g is a point P in the domain of f, return the image f(P);

* if g:E" → E' is a compatible isogeny, return the composite isogeny f o g: E" → E.

      ? one = ffgen(101, 't)^0;
      ? E = ellinit([6, 53, 85, 32, 34] * one);
      ? P = [84, 71] * one;
      ? ellorder(E, P)
      %4 = 5
      ? [F, f] = ellisogeny(E, P);  \\ f: E->F = E/<P>
      ? ellisogenyapply(f, P)
      %6 = [0]
      ? F = ellinit(F);
      ? Q = [89, 44] * one;
      ? ellorder(F, Q)
      %9 = 2
      ? [G, g] = ellisogeny(F, Q); \\  g: F->G = F/<Q>
      ? gof = ellisogenyapply(g, f); \\ gof: E -> G

The library syntax is `GEN ellisogenyapply(GEN f, GEN g)`.

#### ellisomat(E, {p = 0}, {fl = 0})

Given an elliptic curve E defined over a number field, compute representatives of the isomorphism classes of elliptic curves ℚ-isogenous to E. The function returns a vector [L,M] where L is a list of triples [Ei, fi, gi], where Ei is an elliptic curve in [a4,a6] form, fi: E → Ei is a rational isogeny, gi: Ei → E is the dual isogeny of fi, and M is the matrix such that Mi,j is the degree of the isogeny between Ei and Ej. Furthermore the first curve E1 is isomorphic to E by f1. If the flag fl = 1, the fi and gi are not computed, which saves time, and L is the list of the curves Ei. If p is set, it must be a prime number; in this which case only isogenies of degree a power of p are considered. Over a number field, the possible isogeny degrees are determined by Billerey algorithm. As a consequence, CM curves over a number field are not fully supported.

      ? E = ellinit("14a1");
      ? [L,M] = ellisomat(E);
      ? LE = apply(x->x[1], L)  \\ list of curves
      %3 = [[215/48,-5291/864],[-675/16,6831/32],[-8185/48,-742643/864],
           [-1705/48,-57707/864],[-13635/16,306207/32],[-131065/48,-47449331/864]]
      ? L[2][2]  \\ isogeny f2
      %4 = [x^3+3/4*x^2+19/2*x-311/12,
            1/2*x^4+(y+1)*x^3+(y-4)*x^2+(-9*y+23)*x+(55*y+55/2),x+1/3]
      ? L[2][3]  \\ dual isogeny g2
      %5 = [1/9*x^3-1/4*x^2-141/16*x+5613/64,
            -1/18*x^4+(1/27*y-1/3)*x^3+(-1/12*y+87/16)*x^2+(49/16*y-48)*x
            +(-3601/64*y+16947/512),x-3/4]
      ? apply(E->ellidentify(ellinit(E))[1][1], LE)
      %6 = ["14a1","14a4","14a3","14a2","14a6","14a5"]
      ? M
      %7 =
      [1  3  3 2  6  6]

      [3  1  9 6  2 18]

      [3  9  1 6 18  2]

      [2  6  6 1  3  3]

      [6  2 18 3  1  9]

      [6 18  2 3  9  1]

The library syntax is `GEN ellisomat(GEN E, long p, long fl)`.

#### ellisoncurve(E, z)

Gives 1 (i.e. true) if the point z is on the elliptic curve E, 0 otherwise. If E or z have imprecise coefficients, an attempt is made to take this into account, i.e. an imprecise equality is checked, not a precise one. It is allowed for z to be a vector of points in which case a vector (of the same type) is returned.

The library syntax is `GEN ellisoncurve(GEN E, GEN z)`. Also available is `int oncurve(GEN E, GEN z)` which does not accept vectors of points.

#### ellisotree(E)

Given an elliptic curve E defined over ℚ or a set of ℚ-isogenous curves as given by `ellisomat`, return a pair [L,M] where

* L lists the minimal models of the isomorphism classes of elliptic curves ℚ-isogenous to E (or in the set of isogenous curves),

* M is the adjacency matrix of the prime degree isogenies tree: there is an edge from Ei to Ej if there is an isogeny Ei → Ej of prime degree such that the Néron differential forms are preserved.

      ? E = ellinit("14a1");
      ? [L,M] = ellisotree(E);
      ? M
      %3 =
      [0 0 3 2 0 0]

      [3 0 0 0 2 0]

      [0 0 0 0 0 2]

      [0 0 0 0 0 3]

      [0 0 0 3 0 0]

      [0 0 0 0 0 0]
      ? [L2,M2] = ellisotree(ellisomat(E,2,1));
      %4 =
      [0 2]

      [0 0]
      ? [L3,M3] = ellisotree(ellisomat(E,3,1));
      ? M3
      %6 =
      [0 0 3]

      [3 0 0]

      [0 0 0]

Compare with the result of `ellisomat`.

      ? [L,M]=ellisomat(E,,1);
      ? M
      %7 =
      [1  3  3 2  6  6]

      [3  1  9 6  2 18]

      [3  9  1 6 18  2]

      [2  6  6 1  3  3]

      [6  2 18 3  1  9]

      [6 18  2 3  9  1]

The library syntax is `GEN ellisotree(GEN E)`.

#### ellissupersingular(E, {p})

Return 1 if the elliptic curve E defined over a number field, ℚp or a finite field is supersingular at p, and 0 otherwise. If the curve is defined over a number field, p must be explicitly given, and must be a prime number, resp. a maximal ideal, if the curve is defined over ℚ, resp. a general number field: we return 1 if and only if E has supersingular good reduction at p.

Alternatively, E can be given by its j-invariant in a finite field. In this case p must be omitted.

      ? setrand(1); \\ make the choice of g deterministic
      ? g = ffprimroot(ffgen(7^5))
      %1 = 4*x^4 + 5*x^3 + 6*x^2 + 5*x + 6
      ? [g^n | n <- [1 .. 7^5 - 1], ellissupersingular(g^n)]
      %2 = [6]

      ? K = nfinit(y^3-2); P = idealprimedec(K, 2)[1];
      ? E = ellinit([y,1], K);
      ? ellissupersingular(E, P)
      %5 = 1
      ? Q = idealprimedec(K,5)[1];
      ? ellissupersingular(E, Q)
      %6 = 0

The library syntax is `GEN ellissupersingular(GEN E, GEN p = NULL)`. Also available is `int elljissupersingular(GEN j)` where j is a j-invariant of a curve over a finite field.

#### ellj(x)

Elliptic j-invariant. x must be a complex number with positive imaginary part, or convertible into a power series or a p-adic number with positive valuation.

The library syntax is `GEN jell(GEN x, long prec)`.

#### elllocalred(E, {p})

Calculates the Kodaira type of the local fiber of the elliptic curve E at p. E must be an `ell` structure as output by `ellinit`, over ℚ_ℓ (p better left omitted, else equal to ℓ) over ℚ (p a rational prime) or a number field K (p a maximal ideal given by a `prid` structure). The result is a 4-component vector [f,kod,v,c]. Here f is the exponent of p in the arithmetic conductor of E, and kod is the Kodaira type which is coded as follows:

1 means good reduction (type I0), 2, 3 and 4 mean types II, III and IV respectively, 4+ν with ν > 0 means type I_ν; finally the opposite values -1, -2, etc. refer to the starred types I0*, II*, etc. The third component v is itself a vector [u,r,s,t] giving the coordinate changes done during the local reduction; u = 1 if and only if the given equation was already minimal at p. Finally, the last component c is the local Tamagawa number cp.

The library syntax is `GEN elllocalred(GEN E, GEN p = NULL)`.

#### elllog(E, P, G, {o})

Given two points P and G on the elliptic curve E/𝔽q, returns the discrete logarithm of P in base G, i.e. the smallest non-negative integer n such that P = [n]G. See `znlog` for the limitations of the underlying discrete log algorithms. If present, o represents the order of G, see Section se:DLfun; the preferred format for this parameter is `[N, factor(N)]`, where N is the order of G.

If no o is given, assume that G generates the curve. The function also assumes that P is a multiple of G.

      ? a = ffgen(ffinit(2,8),'a);
      ? E = ellinit([a,1,0,0,1]);  \\ over F2^8
      ? x = a^3; y = ellordinate(E,x)[1];
      ? P = [x,y]; G = ellmul(E, P, 113);
      ? ord = [242, factor(242)]; \\ P generates a group of order 242. Initialize.
      ? ellorder(E, G, ord)
      %4 = 242
      ? e = elllog(E, P, G, ord)
      %5 = 15
      ? ellmul(E,G,e) == P
      %6 = 1

The library syntax is `GEN elllog(GEN E, GEN P, GEN G, GEN o = NULL)`.

#### elllseries(E, s, {A = 1})

This function is deprecated, use `lfun(E,s)` instead.

E being an elliptic curve, given by an arbitrary model over ℚ as output by `ellinit`, this function computes the value of the L-series of E at the (complex) point s. This function uses an O(N1/2) algorithm, where N is the conductor.

The optional parameter A fixes a cutoff point for the integral and is best left omitted; the result must be independent of A, up to `realprecision`, so this allows to check the function's accuracy.

The library syntax is `GEN elllseries(GEN E, GEN s, GEN A = NULL, long prec)`.

#### ellminimaldisc(E)

E being an elliptic curve defined over a number field output by `ellinit`, return the minimal discriminant ideal of E.

The library syntax is `GEN ellminimaldisc(GEN E)`.

#### ellminimalmodel(E, {&v})

Let E be an `ell` structure over a number field K. This function determines whether E admits a global minimal integral model. If so, it returns it and sets v = [u,r,s,t] to the corresponding change of variable: the return value is identical to that of `ellchangecurve(E, v)`.

Else return the (non-principal) Weierstrass class of E, i.e. the class of ∏ 𝔭(v𝔭{Δ - δ𝔭) / 12} where Δ = `E.disc` is the model's discriminant and 𝔭 ^ δ𝔭 is the local minimal discriminant. This function requires either that E be defined over the rational field ℚ (with domain D = 1 in `ellinit`), in which case a global minimal model always exists, or over a number field given by a bnf structure. The Weierstrass class is given in `bnfisprincipal` format, i.e. in terms of the `K.gen` generators.

The resulting model has integral coefficients and is everywhere minimal, the coefficients a1 and a3 are reduced modulo 2 (in terms of the fixed integral basis `K.zk`) and a2 is reduced modulo 3. Over ℚ, we further require that a1 and a3 be 0 or 1, that a2 be 0 or ± 1 and that u > 0 in the change of variable: both the model and the change of variable v are then unique.

      ? e = ellinit([6,6,12,55,233]);  \\ over Q
      ? E = ellminimalmodel(e, &v);
      ? E[1..5]
      %3 = [0, 0, 0, 1, 1]
      ? v
      %4 = [2, -5, -3, 9]

      ? K = bnfinit(a^2-65);  \\ over a non-principal number field
      ? K.cyc
      %2 = [2]
      ? u = Mod(8+a, K.pol);
      ? E = ellinit([1,40*u+1,0,25*u^2,0], K);
      ? ellminimalmodel(E) \\ no global minimal model exists over ZK
      %6 = [1]~

The library syntax is `GEN ellminimalmodel(GEN E, GEN *v = NULL)`.

#### ellminimaltwist(E, {flag = 0})

Let E be an elliptic curve defined over ℚ, return a discriminant D such that the twist of E by D is minimal among all possible quadratic twists, i.e. if flag = 0, its minimal model has minimal discriminant, or if flag = 1, it has minimal conductor.

In the example below, we find a curve with j-invariant 3 and minimal conductor.

      ? E = ellminimalmodel(ellinit(ellfromj(3)));
      ? ellglobalred(E)[1]
      %2 = 357075
      ? D = ellminimaltwist(E,1)
      %3 = -15
      ? E2 = ellminimalmodel(ellinit(elltwist(E,D)));
      ? ellglobalred(E2)[1]
      %5 = 14283

In the example below, flag = 0 and flag = 1 give different results.

      ? E = ellinit([1,0]);
      ? D0 = ellminimaltwist(E,0)
      %7 = 1
      ? D1 = ellminimaltwist(E,1)
      %8 = 8
      ? E0 = ellminimalmodel(ellinit(elltwist(E,D0)));
      ? [E0.disc, ellglobalred(E0)[1]]
      %10 = [-64, 64]
      ? E1 = ellminimalmodel(ellinit(elltwist(E,D1)));
      ? [E1.disc, ellglobalred(E1)[1]]
      %12 = [-4096, 32]

The library syntax is `GEN ellminimaltwist0(GEN E, long flag)`. Also available are `GEN ellminimaltwist(E)` for flag = 0, and `GEN ellminimaltwistcond(E)` for flag = 1.

#### ellmoddegree(e)

e being an elliptic curve defined over ℚ output by `ellinit`, compute the modular degree of e divided by the square of the Manin constant c. It is conjectured that c = 1 for the strong Weil curve in the isogeny class (optimal quotient of J0(N)) and this can be proven using `ellweilcurve` when the conductor N is moderate.

      ? E = ellinit("11a1"); \\ from Cremona table: strong Weil curve and c = 1
      ? [v,smith] = ellweilcurve(E); smith \\ proof of the above
      %2 = [[1, 1], [5, 1], [1, 1/5]]
      ? ellmoddegree(E)
      %3 = 1
      ? [ellidentify(e)[1][1] | e<-v]
      %4 = ["11a1", "11a2", "11a3"]
      ? ellmoddegree(ellinit("11a2"))
      %5 = 5
      ? ellmoddegree(ellinit("11a3"))
      %6 = 1/5

The modular degree of `11a1` is 1 (because `ellweilcurve` or Cremona's table prove that the Manin constant is 1 for this curve); the output of `ellweilcurve` also proves that the Manin constants of `11a2` and `11a3` are 1 and 5 respectively, so the actual modular degree of both `11a2` and `11a3` is 5.

The library syntax is `GEN ellmoddegree(GEN e)`.

#### ellmodulareqn(N, {x}, {y})

Given a prime N < 500, return a vector [P,t] where P(x,y) is a modular equation of level N, i.e. a bivariate polynomial with integer coefficients; t indicates the type of this equation: either canonical (t = 0) or Atkin (t = 1). This function requires the `seadata` package and its only use is to give access to the package contents. See `polmodular` for a more general and more flexible function.

Let j be the j-invariant function. The polynomial P satisfies the functional equation, P(f,j) = P(f | WN, j | WN) = 0 for some modular function f = fN (hand-picked for each fixed N to minimize its size, see below), where WN(τ) = -1 / (N τ) is the Atkin-Lehner involution. These two equations allow to compute the values of the classical modular polynomial ΦN, such that ΦN(j(τ), j(Nτ)) = 0, while being much smaller than the latter. More precisely, we have j(WN(τ)) = j(N τ); the function f is invariant under Γ0(N) and also satisfies

* for Atkin type: f | WN = f;

* for canonical type: let s = 12/gcd(12,N-1), then f | WN = N^s / f. In this case, f has a simple definition: f(τ) = N^s (η(N τ) / η(τ) )2 s, where η is Dedekind's eta function.

The following GP function returns values of the classical modular polynomial by eliminating fN(τ) in the above functional equation, for N ≤ 31 or N ∈ {41,47,59,71}.

      classicaleqn(N, X='X, Y='Y)=
      {
        my([P,t] = ellmodulareqn(N), Q, d);
        if (poldegree(P,'y) > 2, error("level unavailable in classicaleqn"));
        if (t == 0, \\ Canonical
          my(s = 12/gcd(12,N-1));
          Q = 'x^(N+1) * substvec(P,['x,'y],[N^s/'x,Y]);
          d = N^(s*(2*N+1)) * (-1)^(N+1);
        , \\ Atkin
          Q = subst(P,'y,Y);
          d = (X-Y)^(N+1));
        polresultant(subst(P,'y,X), Q) / d;
      }

The library syntax is `GEN ellmodulareqn(long N, long x = -1, long y = -1)` where `x`, `y` are variable numbers.

#### ellmul(E, z, n)

Computes [n]z, where z is a point on the elliptic curve E. The exponent n is in ℤ, or may be a complex quadratic integer if the curve E has complex multiplication by n (if not, an error message is issued).

      ? Ei = ellinit([1,0]); z = [0,0];
      ? ellmul(Ei, z, 10)
      %2 = [0]     \\ unsurprising: z has order 2
      ? ellmul(Ei, z, I)
      %3 = [0, 0]  \\ Ei has complex multiplication by Z[i]
      ? ellmul(Ei, z, quadgen(-4))
      %4 = [0, 0]  \\ an alternative syntax for the same query
      ? Ej  = ellinit([0,1]); z = [-1,0];
      ? ellmul(Ej, z, I)
        ***   at top-level: ellmul(Ej,z,I)
        ***                 ^ —  —  —  — --
        *** ellmul: not a complex multiplication in ellmul.
      ? ellmul(Ej, z, 1+quadgen(-3))
      %6 = [1 - w, 0]

The simple-minded algorithm for the CM case assumes that we are in characteristic 0, and that the quadratic order to which n belongs has small discriminant.

The library syntax is `GEN ellmul(GEN E, GEN z, GEN n)`.

#### ellneg(E, z)

Opposite of the point z on elliptic curve E.

The library syntax is `GEN ellneg(GEN E, GEN z)`.

#### ellnonsingularmultiple(E, P)

Given an elliptic curve E/ℚ (more precisely, a model defined over ℚ of a curve) and a rational point P ∈ E(ℚ), returns the pair [R,n], where n is the least positive integer such that R := [n]P has good reduction at every prime. More precisely, its image in a minimal model is everywhere non-singular.

      ? e = ellinit("57a1"); P = [2,-2];
      ? ellnonsingularmultiple(e, P)
      %2 = [[1, -1], 2]
      ? e = ellinit("396b2"); P = [35, -198];
      ? [R,n] = ellnonsingularmultiple(e, P);
      ? n
      %5 = 12

The library syntax is `GEN ellnonsingularmultiple(GEN E, GEN P)`.

#### ellorder(E, z, {o})

Gives the order of the point z on the elliptic curve E, defined over a finite field or a number field. Return (the impossible value) zero if the point has infinite order.

      ? E = ellinit([-157^2,0]);  \\ the "157-is-congruent" curve
      ? P = [2,2]; ellorder(E, P)
      %2 = 2
      ? P = ellheegner(E); ellorder(E, P) \\ infinite order
      %3 = 0
      ? K = nfinit(polcyclo(11,t)); E=ellinit("11a3", K); T = elltors(E);
      ? ellorder(E, T.gen[1])
      %5 = 25
      ? E = ellinit(ellfromj(ffgen(5^10)));
      ? ellcard(E)
      %7 = 9762580
      ? P = random(E); ellorder(E, P)
      %8 = 4881290
      ? p = 2^160+7; E = ellinit([1,2], p);
      ? N = ellcard(E)
      %9 = 1461501637330902918203686560289225285992592471152
      ? o = [N, factor(N)];
      ? for(i=1,100, ellorder(E,random(E)))
      time = 260 ms.

The parameter o, is now mostly useless, and kept for backward compatibility. If present, it represents a non-zero multiple of the order of z, see Section se:DLfun; the preferred format for this parameter is `[ord, factor(ord)]`, where `ord` is the cardinality of the curve. It is no longer needed since PARI is now able to compute it over large finite fields (was restricted to small prime fields at the time this feature was introduced), and caches the result in E so that it is computed and factored only once. Modifying the last example, we see that including this extra parameter provides no improvement:

      ? o = [N, factor(N)];
      ? for(i=1,100, ellorder(E,random(E),o))
      time = 260 ms.

The library syntax is `GEN ellorder(GEN E, GEN z, GEN o = NULL)`. The obsolete form `GEN orderell(GEN e, GEN z)` should no longer be used.

#### ellordinate(E, x)

Gives a 0, 1 or 2-component vector containing the y-coordinates of the points of the curve E having x as x-coordinate.

The library syntax is `GEN ellordinate(GEN E, GEN x, long prec)`.

#### ellpadicL(E, p, n, {s = 0}, {r = 0}, {D = 1})

Returns the value (or r-th derivative) on a character χ^s of ℤp* of the p-adic L-function of the elliptic curve E/ℚ, twisted by D, given modulo p^n.

Characters. The set of continuous characters of Gal(ℚ(μp oo )/ ℚ) is identified to ℤp* via the cyclotomic character χ with values in ℚp*. Denote by τ:ℤp* → ℤp* the Teichmüller character, with values in the (p-1)-th roots of 1 for p != 2, and {-1,1} for p = 2; finally, let `<`χ`>`= χ τ-1, with values in 1 + 2pℤp. In GP, the continuous character of Gal(ℚ(μp oo )/ ℚ) given by `<`χ`>`s1 τs2 is represented by the pair of integers s = (s1,s2), with s1 ∈ ℤp and s2 mod p-1 for p > 2, (resp. mod 2 for p = 2); s may be also an integer, representing (s,s) or χ^s.

The p-adic L function. The p-adic L function Lp is defined on the set of continuous characters of Gal(ℚ(μp oo )/ ℚ), as ∫ℤp* χ^s d μ for a certain p-adic distribution μ on ℤp*. The derivative is given by Lp(r)(E, χ^s) = ∫ℤp* logp^r(a) χ^s(a) dμ(a). More precisely:

* When E has good supersingular reduction, Lp takes its values in D := H^1dR(E/ℚ) ⨂ _ℚ ℚp and satisfies (1-p-1 F)-2 Lp(E, χ^0) = (L(E,1) / Ω).ω where F is the Frobenius, L(E,1) is the value of the complex L function at 1, ω is the Néron differential and Ω the attached period on E(ℝ). Here, χ^0 represents the trivial character.

The function returns the components of Lp(r)(E,χ^s) in the basis (ω, F ω).

* When E has ordinary good reduction, this method only defines the projection of Lp(E,χ^s) on the α-eigenspace, where α is the unit eigenvalue for F. This is what the function returns. We have (1- α-1)-2 Lp,α(E,χ^0) = L(E,1) / Ω.

Two supersingular examples:

      ? cxL(e) = bestappr( ellL1(e) / e.omega[1] );

      ? e = ellinit("17a1"); p=3; \\ supersingular, a3 = 0
      ? L = ellpadicL(e,p,4);
      ? F = [0,-p;1,ellap(e,p)]; \\ Frobenius matrix in the basis (omega,F(omega))
      ? (1-p^(-1)*F)^-2 * L / cxL(e)
      %5 = [1 + O(3^5), O(3^5)]~ \\ [1,0]~

      ? e = ellinit("116a1"); p=3; \\ supersingular, a3 != 0~
      ? L = ellpadicL(e,p,4);
      ? F = [0,-p; 1,ellap(e,p)];
      ? (1-p^(-1)*F)^-2*L~ / cxL(e)
      %9 = [1 + O(3^4), O(3^5)]~

Good ordinary reduction:

      ? e = ellinit("17a1"); p=5; ap = ellap(e,p)
      %1 = -2 \\ ordinary
      ? L = ellpadicL(e,p,4)
      %2 = 4 + 3*5 + 4*5^2 + 2*5^3 + O(5^4)
      ? al = padicappr(x^2 - ap*x + p, ap + O(p^7))[1];
      ? (1-al^(-1))^(-2) * L / cxL(e)
      %4 = 1 + O(5^4)

Twist and Teichmüller:

      ? e = ellinit("17a1"); p=5; \\ ordinary
      \\ 2nd derivative at tau^1, twist by -7
      ? ellpadicL(e, p, 4, [0,1], 2, -7)
      %2 = 2*5^2 + 5^3 + O(5^4)

We give an example of non split multiplicative reduction (see `ellpadicbsd` for more examples).

      ? e=ellinit("15a1"); p=3; n=5;
      ? L = ellpadicL(e,p,n)
      %2 = 2 + 3 + 3^2 + 3^3 + 3^4 + O(3^5)
      ? (1 - ellap(e,p))^(-1) * L / cxL(e)
      %3 = 1 + O(3^5)

This function is a special case of `mspadicL` and it also appears as the first term of `mspadicseries`:

      ? e = ellinit("17a1"); p=5;
      ? L = ellpadicL(e,p,4)
      %2 = 4 + 3*5 + 4*5^2 + 2*5^3 + O(5^4)
      ? [M,phi] = msfromell(e, 1);
      ? Mp = mspadicinit(M, p, 4);
      ? mu = mspadicmoments(Mp, phi);
      ? mspadicL(mu)
      %6 = 4 + 3*5 + 4*5^2 + 2*5^3 + 2*5^4 + 5^5 + O(5^6)
      ? mspadicseries(mu)
      %7 = (4 + 3*5 + 4*5^2 + 2*5^3 + 2*5^4 + 5^5 + O(5^6))
            + (3 + 3*5 + 5^2 + 5^3 + O(5^4))*x
            + (2 + 3*5 + 5^2 + O(5^3))*x^2
            + (3 + 4*5 + 4*5^2 + O(5^3))*x^3
            + (3 + 2*5 + O(5^2))*x^4 + O(x^5)

These are more cumbersome than `ellpadicL` but allow to compute at different characters, or successive derivatives, or to twist by a quadratic character essentially for the cost of a single call to `ellpadicL` due to precomputations.

The library syntax is `GEN ellpadicL(GEN E, GEN p, long n, GEN s = NULL, long r, GEN D = NULL)`.

#### ellpadicbsd(E, p, n, {D = 1})

Given an elliptic curve E over ℚ, its quadratic twist ED and a prime number p, this function is a p-adic analog of the complex functions `ellanalyticrank` and `ellbsd`. It calls `ellpadicL` with initial accuracy p^n and may increase it internally; it returns a vector [r, Lp] where

* Lp is a p-adic number (resp. a pair of p-adic numbers if E has good supersingular reduction) defined modulo p^N, conjecturally equal to Rp S, where Rp is the p-adic regulator as given by `ellpadicregulator` (in the basis (ω, F ω)) and S is the cardinal of the Tate-Shafarevich group for the quadratic twist ED.

* r is an upper bound for the analytic rank of the p-adic L-function attached to ED: we know for sure that the i-th derivative of Lp(ED,.) at χ^0 is O(p^N) for all i < r and that its r-th derivative is non-zero; it is expected that the true analytic rank is equal to the rank of the Mordell-Weil group ED(ℚ), plus 1 if the reduction of ED at p is split multiplicative; if r = 0, then both the analytic rank and the Mordell-Weil rank are unconditionnally 0.

Recall that the p-adic BSD conjecture (Mazur, Tate, Teitelbaum, Bernardi, Perrin-Riou) predicts an explicit link between Rp S and (1-p-1 F)-2.Lp(r)(ED, χ^0) / r! where r is the analytic rank of the p-adic L-function attached to ED and F is the Frobenius on H^1dR; see `ellpadicL` for definitions.

      ? E = ellinit("11a1"); p = 7; n = 5; \\ good ordinary
      ? ellpadicbsd(E, 7, 5) \\ rank 0,
      %2 = [0, 1 + O(7^5)]

      ? E = ellinit("91a1"); p = 7; n = 5; \\ non split multiplicative
      ? [r,Lp] = ellpadicbsd(E, p, n)
      %5 = [1, 2*7 + 6*7^2 + 3*7^3 + 7^4 + O(7^5)]
      ? R = ellpadicregulator(E, p, n, E.gen)
      %6 = 2*7 + 6*7^2 + 3*7^3 + 7^4 + 5*7^5 + O(7^6)
      ? sha = Lp/R
      %7 = 1 + O(7^4)

      ? E = ellinit("91b1"); p = 7; n = 5; \\ split multiplicative
      ? [r,Lp] = ellpadicbsd(E, p, n)
      %9 = [2, 2*7 + 7^2 + 5*7^3 + O(7^4)]
      ? ellpadicregulator(E, p, n, E.gen)
      %10 = 2*7 + 7^2 + 5*7^3 + 6*7^4 + 2*7^5 + O(7^6)
      ? [rC, LC] = ellanalyticrank(E);
      ? [r, rC]
      %12 = [2, 1]  \\ r = rC+1 because of split multiplicative reduction

      ? E = ellinit("53a1"); p = 5; n = 5; \\ supersingular
      ? [r, Lp] = ellpadicbsd(E, p, n);
      ? r
      %15 = 1
      ? Lp
      %16 = [3*5 + 2*5^2 + 2*5^5 + O(5^6), \
             5 + 3*5^2 + 4*5^3 + 2*5^4 + 5^5 + O(5^6)]
      ? R = ellpadicregulator(E, p, n, E.gen)
      %17 = [3*5 + 2*5^2 + 2*5^5 + O(5^6), 5 + 3*5^2 + 4*5^3 + 2*5^4 + O(5^5)]
      \\ expect Lp = R*#Sha, hence (conjecturally) #Sha = 1

      ? E = ellinit("84a1"); p = 11; n = 6; D = -443;
      ? [r,Lp] = ellpadicbsd(E, 11, 6, D) \\ Mordell-Weil rank 0, no regulator
      %19 = [0, 3 + 2*11 + O(11^6)]
      ? lift(Lp)  \\ expected cardinal for Sha is 5^2
      %20 = 25
      ? ellpadicbsd(E, 3, 12, D)  \\ at 3
      %21 = [1, 1 + 2*3 + 2*3^2 + O(3^8)]
      ? ellpadicbsd(E, 7, 8, D)   \\ and at 7
      %22 = [0, 4 + 3*7 + O(7^8)]

The library syntax is `GEN ellpadicbsd(GEN E, GEN p, long n, GEN D = NULL)`.

#### ellpadicfrobenius(E, p, n)

If p > 2 is a prime and E is an elliptic curve on ℚ with good reduction at p, return the matrix of the Frobenius endomorphism ϕ on the crystalline module Dp(E) = ℚp ⨂ H^1dR(E/ℚ) with respect to the basis of the given model (ω, η = x ω), where ω = dx/(2 y+a1 x+a3) is the invariant differential. The characteristic polynomial of ϕ is x^2 - ap x + p. The matrix is computed to absolute p-adic precision p^n.

      ? E = ellinit([1,-1,1,0,0]);
      ? F = ellpadicfrobenius(E,5,3);
      ? lift(F)
      %3 =
      [120 29]

      [ 55  5]
      ? charpoly(F)
      %4 = x^2 + O(5^3)*x + (5 + O(5^3))
      ? ellap(E, 5)
      %5 = 0

The library syntax is `GEN ellpadicfrobenius(GEN E, long p, long n)`.

#### ellpadicheight(E, p, n, P, {Q})

Cyclotomic p-adic height of the rational point P on the elliptic curve E (defined over ℚ), given to n p-adic digits. If the argument Q is present, computes the value of the bilinear form (h(P+Q)-h(P-Q)) / 4.

Let D := H^1dR(E) ⨂ _ℚ ℚp be the ℚp vector space spanned by ω (invariant differential dx/(2y+a_1x+a3) related to the given model) and η = x ω. Then the cyclotomic p-adic height hE associates to P ∈ E(ℚ) an element f ω + g η in D. This routine returns the vector [f, g] to n p-adic digits. If P ∈ E(ℚ) is in the kernel of reduction mod p and if its reduction at all finite places is non singular, then g = -(logE P)^2, where logE is the logarithm for the formal group of E at p.

If furthermore the model is of the form Y^2 = X^3 + a X + b and P = (x,y), then f = logp(`denominator`(x)) - 2 logp(σ(P)) where σ(P) is given by `ellsigma`(E,P).

Recall (Advanced topics in the arithmetic of elliptic curves, Theorem 3.2) that the local height function over the complex numbers is of the form λ(z) = -log (|`E.disc`|) / 6 + Re(z η(z)) - 2 log( σ(z)). (N.B. our normalization for local and global heights is twice that of Silverman's).

       ? E = ellinit([1,-1,1,0,0]); P = [0,0];
       ? ellpadicheight(E,5,3, P)
       %2 = [3*5 + 5^2 + 2*5^3 + O(5^4), 5^2 + 4*5^4 + O(5^5)]
       ? E = ellinit("11a1"); P = [5,5]; \\ torsion point
       ? ellpadicheight(E,19,6, P)
       %4 = [0, 0]
       ? E = ellinit([0,0,1,-4,2]); P = [-2,1];
       ? ellpadicheight(E,3,3, P)
       %6 = [2*3^2 + 2*3^3 + 3^4 + O(3^5), 2*3^2 + 3^4 + O(3^5)]
       ? ellpadicheight(E,3,5, P, elladd(E,P,P))
       %7 = [3^2 + 2*3^3 + O(3^7), 3^2 + 3^3 + 2*3^4 + 3^5 + O(3^7)]

* When E has good ordinary reduction at p or non split multiplicative reduction, the "canonical" p-adic height is given by

      s2 = ellpadics2(E,p,n);
      ellpadicheight(E, p, n, P) * [1,-s2]~

Since s2 does not depend on P, it is preferable to compute it only once:

      ? E = ellinit("5077a1"); p = 5; n = 7;  \\ rank 3
      ? s2 = ellpadics2(E,p,n);
      ? M = ellpadicheightmatrix(E,p, n, E.gen) * [1,-s2]~;
      ? matdet(M)   \\ p-adic regulator on the points in E.gen
      %4 = 5 + 5^2 + 4*5^3 + 2*5^4 + 2*5^5 + 2*5^6 + O(5^7)

* When E has split multiplicative reduction at p (Tate curve), the "canonical" p-adic height is given by

      Ep = ellinit(E[1..5], O(p^(n))); \\ E seen as a Tate curve over Qp
      [u2,u,q] = Ep.tate;
      ellpadicheight(E, p, n, P) * [1,-s2 + 1/log(q)/u2]]~

where s2 is as above. For example,

      ? E = ellinit("91b1"); P =[-1, 3]; p = 7; n = 5;
      ? Ep = ellinit(E[1..5], O(p^(n)));
      ? s2 = ellpadics2(E,p,n);
      ? [u2,u,q] = Ep.tate;
      ? H = ellpadicheight(E,p, n, P) * [1,-s2 + 1/log(q)/u2]~
      %5 = 2*7 + 7^2 + 5*7^3 + 6*7^4 + 2*7^5 + O(7^6)

These normalizations are chosen so that p-adic BSD conjectures are easy to state, see `ellpadicbsd`.

The library syntax is `GEN ellpadicheight0(GEN E, GEN p, long n, GEN P, GEN Q = NULL)`.

#### ellpadicheightmatrix(E, p, n, Q)

Q being a vector of points, this function returns the "Gram matrix" [F,G] of the cyclotomic p-adic height hE with respect to the basis (ω, η) of D = H^1dR(E) ⨂ _ℚ ℚp given to n p-adic digits. In other words, if `ellpadicheight`(E,p,n, Q[i],Q[j]) = [f,g], corresponding to f ω + g η in D, then F[i,j] = f and G[i,j] = g.

      ? E = ellinit([0,0,1,-7,6]); Q = [[-2,3],[-1,3]]; p = 5; n = 5;
      ? [F,G] = ellpadicheightmatrix(E,p,n,Q);
      ? lift(F)  \\ p-adic entries, integral approximation for readability
      %3 =
      [2364 3100]

      [3100 3119]

      ? G
      %4 =
      [25225 46975]

      [46975 61850]

      ? [F,G] * [1,-ellpadics2(E,p,n)]~
      %5 =
      [4 + 2*5 + 4*5^2 + 3*5^3 + O(5^5)           4*5^2 + 4*5^3 + 5^4 + O(5^5)]

      [    4*5^2 + 4*5^3 + 5^4 + O(5^5) 4 + 3*5 + 4*5^2 + 4*5^3 + 5^4 + O(5^5)]

The library syntax is `GEN ellpadicheightmatrix(GEN E, GEN p, long n, GEN Q)`.

#### ellpadiclog(E, p, n, P)

Given E defined over K = ℚ or ℚp and P = [x,y] on E(K) in the kernel of reduction mod p, let t(P) = -x/y be the formal group parameter; this function returns L(t), where L denotes the formal logarithm (mapping the formal group of E to the additive formal group) attached to the canonical invariant differential: dL = dx/(2y + a_1x + a3).

      ? E = ellinit([0,0,1,-4,2]); P = [-2,1];
      ? ellpadiclog(E,2,10,P)
      %2 = 2 + 2^3 + 2^8 + 2^9 + 2^10 + O(2^11)
      ? E = ellinit([17,42]);
      ? p=3; Ep = ellinit(E,p); \\ E mod p
      ? P=[114,1218]; ellorder(Ep,P) \\ the order of P on (E mod p) is 2
      %5 = 2
      ? Q = ellmul(E,P,2) \\ we need a point of the form 2*P
      %6 = [200257/7056, 90637343/592704]
      ? ellpadiclog(E,3,10,Q)
      %7 = 3 + 2*3^2 + 3^3 + 3^4 + 3^5 + 3^6 + 2*3^8 + 3^9 + 2*3^10 + O(3^11)

The library syntax is `GEN ellpadiclog(GEN E, GEN p, long n, GEN P)`.

#### ellpadicregulator(E, p, n, S)

Let E/ℚ be an elliptic curve. Return the determinant of the Gram matrix of the vector of points S = (S1,..., Sr) with respect to the "canonical" cyclotomic p-adic height on E, given to n (p-adic) digits.

When E has ordinary reduction at p, this is the expected Gram deteterminant in ℚp.

In the case of supersingular reduction of E at p, the definition requires care: the regulator R is an element of D := H^1dR(E) ⨂ _ℚ ℚp, which is a two-dimensional ℚp-vector space spanned by ω and η = x ω (which are defined over ℚ) or equivalently but now over ℚp by ω and Fω where F is the Frobenius endomorphism on D as defined in `ellpadicfrobenius`. On D we define the cyclotomic height hE = f ω + g η (see `ellpadicheight`) and a canonical alternating bilinear form [.,.]D such that [ω, η]D = 1.

For any ν ∈ D, we can define a height h_ν := [ hE, ν ]D from E(ℚ) to ℚp and `<`.,.`>`_ν the attached bilinear form. In particular, if hE = f ω + gη, then h_η = [ hE, η ]D = f and h_ω = [ hE, ω ]D = - g hence hE = h_η ω - h_ω η. Then, R is the unique element of D such that [ω,ν]Dr-1 [R, ν]D = det(`<`Si, Sj `>`ν) for all ν ∈ D not in ℚp ω. The `ellpadicregulator` function returns R in the basis (ω, Fω), which was chosen so that p-adic BSD conjectures are easy to state, see `ellpadicbsd`.

Note that by definition [R, η]D = det(`<`Si, Sj `>`η) and [R, ω+η]D = det(`<`Si, Sj `>`ω+η).

The library syntax is `GEN ellpadicregulator(GEN E, GEN p, long n, GEN S)`.

#### ellpadics2(E, p, n)

If p > 2 is a prime and E/ℚ is an elliptic curve with ordinary good reduction at p, returns the slope of the unit eigenvector of `ellpadicfrobenius(E,p,n)`, i.e., the action of Frobenius ϕ on the crystalline module Dp(E) = ℚp ⨂ H^1dR(E/ℚ) in the basis of the given model (ω, η = x ω), where ω is the invariant differential dx/(2 y+a1 x+a3). In other words, η + s2ω is an eigenvector for the unit eigenvalue of ϕ.

      ? e=ellinit([17,42]);
      ? ellpadics2(e,13,4)
      %2 = 10 + 2*13 + 6*13^3 + O(13^4)

This slope is the unique c ∈ 3-1ℤp such that the odd solution σ(t) = t + O(t^2) of - d((1)/(σ) (d σ)/(ω)) = (x(t) + c) ω is in tℤp[[t]].

It is equal to b2/12 - E2/12 where E2 is the value of the Katz p-adic Eisenstein series of weight 2 on (E,ω). This is used to construct a canonical p-adic height when E has good ordinary reduction at p as follows

      s2 = ellpadics2(E,p,n);
      h(E,p,n, P, s2) = ellpadicheight(E, [p,[1,-s2]],n, P);

Since s2 does not depend on the point P, we compute it only once.

The library syntax is `GEN ellpadics2(GEN E, GEN p, long n)`.

#### ellperiods(w, {flag = 0})

Let w describe a complex period lattice (w = [w1,w2] or an `ellinit` structure). Returns normalized periods [W1,W2] generating the same lattice such that τ := W1/W2 has positive imaginary part and lies in the standard fundamental domain for SL2(ℤ).

If flag = 1, the function returns [[W1,W2], [η1,η2]], where η1 and η2 are the quasi-periods attached to [W1,W2], satisfying η2 W1 - η1 W2 = 2 i π.

The output of this function is meant to be used as the first argument given to ellwp, ellzeta, ellsigma or elleisnum. Quasi-periods are needed by ellzeta and ellsigma only.

      ? L = ellperiods([1,I],1);
      ? [w1,w2] = L[1]; [e1,e2] = L[2];
      ? e2*w1 - e1*w2
      %3 = 6.2831853071795864769252867665590057684*I
      ? ellzeta(L, 1/2 + 2*I)
      %4 = 1.5707963... - 6.283185307...*I
      ? ellzeta([1,I], 1/2 + 2*I) \\ same but less efficient
      %4 = 1.5707963... - 6.283185307...*I

The library syntax is `GEN ellperiods(GEN w, long flag, long prec)`.

#### ellpointtoz(E, P)

If E/ℂ ~ ℂ/Λ is a complex elliptic curve (Λ = `E.omega`), computes a complex number z, well-defined modulo the lattice Λ, corresponding to the point P; i.e. such that P = [℘_Λ(z),℘'_Λ(z)] satisfies the equation y^2 = 4x^3 - g2 x - g3, where g2, g3 are the elliptic invariants.

If E is defined over ℝ and P ∈ E(ℝ), we have more precisely, 0 ≤ Re(t) < w1 and 0 ≤ Im(t) < Im(w2), where (w1,w2) are the real and complex periods of E.

      ? E = ellinit([0,1]); P = [2,3];
      ? z = ellpointtoz(E, P)
      %2 = 3.5054552633136356529375476976257353387
      ? ellwp(E, z)
      %3 = 2.0000000000000000000000000000000000000
      ? ellztopoint(E, z) - P
      %4 = [2.548947057811923643 E-57, 7.646841173435770930 E-57]
      ? ellpointtoz(E, [0]) \\ the point at infinity
      %5 = 0

If E is defined over a general number field, the function returns the values corresponding to the various complex embeddings of the curve and of the point, in the same order as `E.nf.roots`:

      ? E=ellinit([-22032-15552*x,0], nfinit(x^2-2));
      ? P=[-72*x-108,0];
      ? ellisoncurve(E,P)
      %3 = 1
      ? ellpointtoz(E,P)
      %4 = [-0.52751724240790530394437835702346995884*I,
            -0.090507650025885335533571758708283389896*I]
      ? E.nf.roots
      %5 = [-1.4142135623730950488016887242096980786, \\ x-> -sqrt(2)
             1.4142135623730950488016887242096980786] \\ x->  sqrt(2)

If E/ℚp has multiplicative reduction, then E/ℚp is analytically isomorphic to ℚp*/q^ℤ (Tate curve) for some p-adic integer q. The behavior is then as follows:

* If the reduction is split (E.`tate[2]` is a `t_PADIC`), we have an isomorphism φ: E(ℚp) ~ ℚp*/q^ℤ and the function returns φ(P) ∈ ℚp.

* If the reduction is not split (E.`tate[2]` is a `t_POLMOD`), we only have an isomorphism φ: E(K) ~ K*/q^ℤ over the unramified quadratic extension K/ℚp. In this case, the output φ(P) ∈ K is a `t_POLMOD`.

      ? E = ellinit([0,-1,1,0,0], O(11^5)); P = [0,0];
      ? [u2,u,q] = E.tate; type(u) \\ split multiplicative reduction
      %2 = "t_PADIC"
      ? ellmul(E, P, 5)  \\ P has order 5
      %3 = [0]
      ? z = ellpointtoz(E, [0,0])
      %4 = 3 + 11^2 + 2*11^3 + 3*11^4 + 6*11^5 + 10*11^6 + 8*11^7 + O(11^8)
      ? z^5
      %5 = 1 + O(11^9)
      ? E = ellinit(ellfromj(1/4), O(2^6)); x=1/2; y=ellordinate(E,x)[1];
      ? z = ellpointtoz(E,[x,y]); \\ t_POLMOD of t_POL with t_PADIC coeffs
      ? liftint(z) \\ lift all p-adics
      %8 = Mod(8*u + 7, u^2 + 437)

The library syntax is `GEN zell(GEN E, GEN P, long prec)`.

#### ellpow(E, z, n)

Deprecated alias for `ellmul`.

The library syntax is `GEN ellmul(GEN E, GEN z, GEN n)`.

#### ellratpoints(E, h, {flag = 0})

E being an integral model of elliptic curve , return a vector containing the affine rational points on the curve of naive height less than h. If flag = 1, stop as soon as a point is found; return either an empty vector or a vector containing a single point. See `hyperellratpoints` for how h can be specified.

      ? E=ellinit([-25,1]);
      ? ellratpoints(E,10)
      %2 = [[-5,1],[-5,-1],[-3,7],[-3,-7],[-1,5],[-1,-5],
            [0,1],[0,-1],[5,1],[5,-1],[7,13],[7,-13]]
      ? ellratpoints(E,10,1)
      %3 = [[-5,1]]

The library syntax is `GEN ellratpoints(GEN E, GEN h, long flag)`.

#### ellrootno(E, {p})

E being an `ell` structure over ℚ as output by `ellinit`, this function computes the local root number of its L-series at the place p (at the infinite place if p = 0). If p is omitted, return the global root number and in this case the curve can also be defined over a number field.

Note that the global root number is the sign of the functional equation and conjecturally is the parity of the rank of the Mordell-Weil group. The equation for E needs not be minimal at p, but if the model is already minimal the function will run faster.

The library syntax is `long ellrootno(GEN E, GEN p = NULL)`.

#### ellsea(E, {tors = 0})

Let E be an ell structure as output by `ellinit`, defined over a finite field 𝔽q. This low-level function computes the order of the group E(𝔽q) using the SEA algorithm; compared to the high-level function `ellcard`, which includes SEA among its choice of algorithms, the `tors` argument allows to speed up a search for curves having almost prime order and whose quadratic twist may also have almost prime order. When `tors` is set to a non-zero value, the function returns 0 as soon as it detects that the order has a small prime factor not dividing `tors`; SEA considers modular polynomials of increasing prime degree ℓ and we return 0 as soon as we hit an ℓ (coprime to `tors`) dividing #E(𝔽q):

      ? ellsea(ellinit([1,1], 2^56+3477), 1)
      %1 = 72057594135613381
      ? forprime(p=2^128,oo, q = ellcard(ellinit([1,1],p)); if(isprime(q),break))
      time = 6,571 ms.
      ? forprime(p=2^128,oo, q = ellsea(ellinit([1,1],p),1);if(isprime(q),break))
      time = 522 ms.

In particular, set `tors` to 1 if you want a curve with prime order, to 2 if you want to allow a cofactor which is a power of two (e.g. for Edwards's curves), etc. The early exit on bad curves yields a massive speedup compared to running the cardinal algorithm to completion.

When `tors` is negative, similar checks are performed for the quadratic twist of the curve.

The following function returns a curve of prime order over 𝔽p.

      cryptocurve(p) =
      {
        while(1,
          my(E, N, j = Mod(random(p), p));
          E = ellinit(ellfromj(j));
          N = ellsea(E, 1); if (!N, continue);
          if (isprime(N), return(E));
          \\ try the quadratic twist for free
          if (isprime(2*p+2 - N), return(ellinit(elltwist(E))));
        );
      }
      ? p = randomprime([2^255, 2^256]);
      ? E = cryptocurve(p); \\ insist on prime order
      %2 = 47,447ms

The same example without early abort (using `ellcard(E)` instead of `ellsea(E, 1)`) runs for about 5 minutes before finding a suitable curve.

The availability of the `seadata` package will speed up the computation, and is strongly recommended. The generic function `ellcard` should be preferred when you only want to compute the cardinal of a given curve without caring about it having almost prime order:

* If the characteristic is too small (p ≤ 7) or the field cardinality is tiny (q ≤ 523) the generic algorithm `ellcard` is used instead and the `tors` argument is ignored. (The reason for this is that SEA is not implemented for p ≤ 7 and that if q ≤ 523 it is likely to run into an infinite loop.)

* If the field cardinality is smaller than about 250, the generic algorithm will be faster.

* Contrary to `ellcard`, `ellsea` does not store the computed cardinality in E.

The library syntax is `GEN ellsea(GEN E, long tors)`.

#### ellsearch(N)

This function finds all curves in the `elldata` database satisfying the constraint defined by the argument N:

* if N is a character string, it selects a given curve, e.g. `"11a1"`, or curves in the given isogeny class, e.g. `"11a"`, or curves with given conductor, e.g. `"11"`;

* if N is a vector of integers, it encodes the same constraints as the character string above, according to the `ellconvertname` correspondance, e.g. `[11,0,1]` for `"11a1"`, `[11,0]` for `"11a"` and `[11]` for `"11"`;

* if N is an integer, curves with conductor N are selected.

If N codes a full curve name, for instance `"11a1"` or `[11,0,1]`, the output format is [N, [a1,a2,a3,a4,a6], G] where [a1,a2,a3,a4,a6] are the coefficients of the Weierstrass equation of the curve and G is a ℤ-basis of the free part of the Mordell-Weil group attached to the curve.

      ? ellsearch("11a3")
      %1 = ["11a3", [0, -1, 1, 0, 0], []]
      ? ellsearch([11,0,3])
      %2 = ["11a3", [0, -1, 1, 0, 0], []]

If N is not a full curve name, then the output is a vector of all matching curves in the above format:

      ? ellsearch("11a")
      %1 = [["11a1", [0, -1, 1, -10, -20], []],
            ["11a2", [0, -1, 1, -7820, -263580], []],
            ["11a3", [0, -1, 1, 0, 0], []]]
      ? ellsearch("11b")
      %2 = []

The library syntax is `GEN ellsearch(GEN N)`. Also available is `GEN ellsearchcurve(GEN N)` that only accepts complete curve names (as `t_STR`).

#### ellsigma(L, {z = 'x}, {flag = 0})

Computes the value at z of the Weierstrass σ function attached to the lattice L as given by `ellperiods`(,1): including quasi-periods is useful, otherwise there are recomputed from scratch for each new z. σ(z, L) = z ∏ω ∈ L* (1 - (z)/(ω))e(z)/(ω) + (z^2)/(2ω^2). It is also possible to directly input L = [ω1,ω2], or an elliptic curve E as given by `ellinit` (L = `E.omega`).

      ? w = ellperiods([1,I], 1);
      ? ellsigma(w, 1/2)
      %2 = 0.47494937998792065033250463632798296855
      ? E = ellinit([1,0]);
      ? ellsigma(E) \\ at 'x, implicitly at default seriesprecision
      %4 = x + 1/60*x^5 - 1/10080*x^9 - 23/259459200*x^13 + O(x^17)

If flag = 1, computes an arbitrary determination of log(σ(z)).

The library syntax is `GEN ellsigma(GEN L, GEN z = NULL, long flag, long prec)`.

#### ellsub(E, z1, z2)

Difference of the points z1 and z2 on the elliptic curve corresponding to E.

The library syntax is `GEN ellsub(GEN E, GEN z1, GEN z2)`.

#### elltamagawa(E)

The object E being an elliptic curve over a number field, returns the global Tamagawa number of the curve (including the factor at infinite places).

      ? e = ellinit([1, -1, 1, -3002, 63929]); \\ curve "90c6" from elldata
      ? elltamagawa(e)
      %2 = 288
      ? [elllocalred(e,p)[4] | p<-[2,3,5]]
      %3 = [6, 4, 6]
      ? vecprod(%)  \\ since e.disc > 0 the factor at infinity is 2
      %4 = 144

The library syntax is `GEN elltamagawa(GEN E)`.

#### elltaniyama(E, {d = seriesprecision})

Computes the modular parametrization of the elliptic curve E/ℚ, where E is an `ell` structure as output by `ellinit`. This returns a two-component vector [u,v] of power series, given to d significant terms (`seriesprecision` by default), characterized by the following two properties. First the point (u,v) satisfies the equation of the elliptic curve. Second, let N be the conductor of E and Φ: X0(N) → E be a modular parametrization; the pullback by Φ of the Néron differential du/(2v+a_1u+a3) is equal to 2iπ f(z)dz, a holomorphic differential form. The variable used in the power series for u and v is x, which is implicitly understood to be equal to exp(2iπ z).

The algorithm assumes that E is a strong Weil curve and that the Manin constant is equal to 1: in fact, f(x) = ∑n > 0 `ellan`(E, n) x^n.

The library syntax is `GEN elltaniyama(GEN E, long precdl)`.

#### elltatepairing(E, P, Q, m)

Computes the Tate pairing of the two points P and Q on the elliptic curve E. The point P must be of m-torsion.

The library syntax is `GEN elltatepairing(GEN E, GEN P, GEN Q, GEN m)`.

#### elltors(E)

If E is an elliptic curve defined over a number field or a finite field, outputs the torsion subgroup of E as a 3-component vector `[t,v1,v2]`, where `t` is the order of the torsion group, `v1` gives the structure of the torsion group as a product of cyclic groups (sorted by decreasing order), and `v2` gives generators for these cyclic groups. E must be an `ell` structure as output by `ellinit`.

      ?  E = ellinit([-1,0]);
      ?  elltors(E)
      %1 = [4, [2, 2], [[0, 0], [1, 0]]]

Here, the torsion subgroup is isomorphic to ℤ/2ℤ x ℤ/2ℤ, with generators [0,0] and [1,0].

The library syntax is `GEN elltors(GEN E)`.

#### elltwist(E, {P})

Returns the coefficients [a1,a2,a3,a4,a6] of the twist of the elliptic curve E by the quadratic extension of the coefficient ring defined by P (when P is a polynomial) or `quadpoly(P)` when P is an integer. If E is defined over a finite field, then P can be omitted, in which case a random model of the unique non-trivial twist is returned. If E is defined over a number field, the model should be replaced by a minimal model (if one exists).

Example: Twist by discriminant -3:

      ? elltwist(ellinit([0,a2,0,a4,a6]),-3)
      %1 = [0,-3*a2,0,9*a4,-27*a6]

Twist by the Artin-Shreier extension given by x^2+x+T in characteristic 2:

      ? lift(elltwist(ellinit([a1,a2,a3,a4,a6]*Mod(1,2)),x^2+x+T))
      %1 = [a1,a2+a1^2*T,a3,a4,a6+a3^2*T]

Twist of an elliptic curve defined over a finite field:

      ? E=ellinit([1,7]*Mod(1,19));lift(elltwist(E))
      %1 = [0,0,0,11,12]

The library syntax is `GEN elltwist(GEN E, GEN P = NULL)`.

#### ellweilcurve(E, {&ms})

If E' is an elliptic curve over ℚ, let LE' be the sub-ℤ-module of HomΓ0(N)(Δ,ℚ) attached to E' (It is given by x[3] if [M,x] = `msfromell`(E').)

On the other hand, if N is the conductor of E and f is the modular form for Γ0(N) attached to E, let Lf be the lattice of the f-component of HomΓ0(N)(Δ,ℚ) given by the elements φ such that φ({0,γ-1 0}) ∈ ℤ for all γ ∈ Γ0(N) (see `mslattice`).

Let E' run through the isomorphism classes of elliptic curves isogenous to E as given by `ellisomat` (and in the same order). This function returns a pair `[vE,vS]` where `vE` contains minimal models for the E' and `vS` contains the list of Smith invariants for the lattices LE' in Lf. The function also accepts the output of `ellisomat`, i.e. the isogeny class. If the optional argument `ms` is present, it contains the output of `msfromell(vE, 0)`, i.e. the new modular symbol space M of level N and a vector of triples [x^+,x^-, L] attached to each curve E'.

In particular, the strong Weil curve amongst the curves isogenous to E is the one whose Smith invariants are [c,c], where c is the Manin constant, conjecturally equal to 1.

      ? E = ellinit("11a3");
      ? [vE, vS] = ellweilcurve(E);
      ? [n] = [ i | i<-[1..#vS], vS[i]==[1,1] ]  \\ lattice with invariant [1,1]
      %3 = [2]
      ? ellidentify(vE[n]) \\ ... corresponds to strong Weil curve
      %4 = [["11a1", [0, -1, 1, -10, -20], []], [1, 0, 0, 0]]

      ? [vE, vS] = ellweilcurve(E, &ms); \\ vE,vS are as above
      ? [M, vx] = ms; msdim(M) \\ ... but ms contains more information
      %6 = 3
      ? #vx
      %7 = 3
      ? vx[1]
      %8 = [[1/25, -1/10, -1/10]~, [0, 1/2, -1/2]~, [1/25,0; -3/5,1; 2/5,-1]]
      ? forell(E, 11,11, print(msfromell(ellinit(E[1]), 1)[2]))
      [1/5, -1/2, -1/2]~
      [1, -5/2, -5/2]~
      [1/25, -1/10, -1/10]~

The last example prints the modular symbols x^+ in M^+ attached to the curves `11a1`, `11a2` and `11a3`.

The library syntax is `GEN ellweilcurve(GEN E, GEN *ms = NULL)`.

#### ellweilpairing(E, P, Q, m)

Computes the Weil pairing of the two points of m-torsion P and Q on the elliptic curve E.

The library syntax is `GEN ellweilpairing(GEN E, GEN P, GEN Q, GEN m)`.

#### ellwp(w, {z = 'x}, {flag = 0})

Computes the value at z of the Weierstrass ℘ function attached to the lattice w as given by `ellperiods`. It is also possible to directly input w = [ω1,ω2], or an elliptic curve E as given by `ellinit` (w = `E.omega`).

      ? w = ellperiods([1,I]);
      ? ellwp(w, 1/2)
      %2 = 6.8751858180203728274900957798105571978
      ? E = ellinit([1,1]);
      ? ellwp(E, 1/2)
      %4 = 3.9413112427016474646048282462709151389

One can also compute the series expansion around z = 0:

      ? E = ellinit([1,0]);
      ? ellwp(E)              \\ 'x implicitly at default seriesprecision
      %5 = x^-2 - 1/5*x^2 + 1/75*x^6 - 2/4875*x^10 + O(x^14)
      ? ellwp(E, x + O(x^12)) \\ explicit precision
      %6 = x^-2 - 1/5*x^2 + 1/75*x^6 + O(x^9)

Optional flag means 0 (default): compute only ℘(z), 1: compute [℘(z),℘'(z)].

For instance, the Dickson elliptic functions sm and sn can be implemented as follows

       smcm(z) =
       { my(a, b, E = ellinit([0,-1/(4*27)])); \\ ell. invariants (g2,g3)=(0,1/27)
         [a,b] = ellwp(E, z, 1);
         [6*a / (1-3*b), (3*b+1)/(3*b-1)];
       }
       ? [s,c] = smcm(0.5);
       ? s
       %2 = 0.4898258757782682170733218609
       ? c
       %3 = 0.9591820206453842491187464098
       ? s^3+c^3
       %4 = 1.000000000000000000000000000
       ? smcm('x + O('x^11))
       %5 = [x - 1/6*x^4 + 2/63*x^7 - 13/2268*x^10 + O(x^11),
             1 - 1/3*x^3 + 1/18*x^6 - 23/2268*x^9 + O(x^10)]

The library syntax is `GEN ellwp0(GEN w, GEN z = NULL, long flag, long prec)`. For flag = 0, we also have `GEN ellwp(GEN w, GEN z, long prec)`, and `GEN ellwpseries(GEN E, long v, long precdl)` for the power series in variable v.

#### ellxn(E, n, {v = 'x})

In standard notation, for any affine point P = (v,w) on the curve E, we have [n]P = (φn(P)ψn(P) : ωn(P) : ψn(P)^3) for some polynomials φn,ωn,ψn in ℤ[a1,a2,a3,a4,a6][v,w]. This function returns [φn(P),ψn(P)^2], which give the numerator and denominator of the abscissa of [n]P and depend only on v.

      ? E = ellinit([17,42]);
      ? T = ellxn(E, 2, 'X)
      %2 = [X^4 - 34*X^2 - 336*X + 289, 4*X^3 + 68*X + 168]
      ? P = [114,1218]; ellmul(E,P,2)
      %3 = [200257/7056, 90637343/592704]
      ? [x,y] = subst(T,'X,P[1]) \\ substitute P[1] in ellxn(E,2)
      %4 = [168416137, 5934096] \\ numerator and denominator of 2*P
      ? x/y                     \\ check we find ellmul(e,P,2)[1]
      %5 = 200257/7056

The library syntax is `GEN ellxn(GEN E, long n, long v = -1)` where `v` is a variable number.

#### ellzeta(w, {z = 'x})

Computes the value at z of the Weierstrass ζ function attached to the lattice w as given by `ellperiods`(,1): including quasi-periods is useful, otherwise there are recomputed from scratch for each new z. ζ(z, L) = (1)/(z) + z^2∑ω ∈ L* (1)/(ω^2(z-ω)). It is also possible to directly input w = [ω1,ω2], or an elliptic curve E as given by `ellinit` (w = `E.omega`). The quasi-periods of ζ, such that ζ(z + aω1 + bω2) = ζ(z) + aη1 + bη2 for integers a and b are obtained as ηi = 2ζ(ωi/2). Or using directly `elleta`.

      ? w = ellperiods([1,I],1);
      ? ellzeta(w, 1/2)
      %2 = 1.5707963267948966192313216916397514421
      ? E = ellinit([1,0]);
      ? ellzeta(E, E.omega[1]/2)
      %4 = 0.84721308479397908660649912348219163647

One can also compute the series expansion around z = 0 (the quasi-periods are useless in this case):

      ? E = ellinit([0,1]);
      ? ellzeta(E) \\ at 'x, implicitly at default seriesprecision
      %4 = x^-1 + 1/35*x^5 - 1/7007*x^11 + O(x^15)
      ? ellzeta(E, x + O(x^20)) \\ explicit precision
      %5 = x^-1 + 1/35*x^5 - 1/7007*x^11 + 1/1440257*x^17 + O(x^18)

The library syntax is `GEN ellzeta(GEN w, GEN z = NULL, long prec)`.

#### ellztopoint(E, z)

E being an ell as output by `ellinit`, computes the coordinates [x,y] on the curve E corresponding to the complex or p-adic parameter z. Hence this is the inverse function of `ellpointtoz`.

* If E is defined over a p-adic field and has multiplicative reduction, then z is understood as an element on the Tate curve Qp* / q^ℤ.

      ? E = ellinit([0,-1,1,0,0], O(11^5));
      ? [u2,u,q] = E.tate; type(u)
      %2 = "t_PADIC" \\ split multiplicative reduction
      ? z = ellpointtoz(E, [0,0])
      %3 = 3 + 11^2 + 2*11^3 + 3*11^4 + 6*11^5 + 10*11^6 + 8*11^7 + O(11^8)
      ? ellztopoint(E,z)
      %4 = [O(11^9), O(11^9)]

      ? E = ellinit(ellfromj(1/4), O(2^6)); x=1/2; y=ellordinate(E,x)[1];
      ? z = ellpointtoz(E,[x,y]); \\ non-split: t_POLMOD with t_PADIC coefficients
      ? P = ellztopoint(E, z);
      ? P[1] \\ y coordinate is analogous, more complicated
      %8 = Mod(O(2^4)*x + (2^-1 + O(2^5)), x^2 + (1 + 2^2 + 2^4 + 2^5 + O(2^7)))

* If E is defined over the complex numbers (for instance over ℚ), z is understood as a complex number in ℂ/ΛE. If the short Weierstrass equation is y^2 = 4x^3 - g_2x - g3, then [x,y] represents the Weierstrass ℘-function and its derivative. For a general Weierstrass equation we have x = ℘(z) - b2/12, y = ℘'(z)/2 - (a1 x + a3)/2. If z is in the lattice defining E over ℂ, the result is the point at infinity [0].

      ? E = ellinit([0,1]); P = [2,3];
      ? z = ellpointtoz(E, P)
      %2 = 3.5054552633136356529375476976257353387
      ? ellwp(E, z)
      %3 = 2.0000000000000000000000000000000000000
      ? ellztopoint(E, z) - P
      %4 = [2.548947057811923643 E-57, 7.646841173435770930 E-57]
      ? ellztopoint(E, 0)
      %5 = [0] \\ point at infinity

The library syntax is `GEN pointell(GEN E, GEN z, long prec)`.

#### genus2red(PQ, {p})

Let PQ be a polynomial P, resp. a vector [P,Q] of polynomials, with rational coefficients. Determines the reduction at p > 2 of the (proper, smooth) genus 2 curve C/ℚ, defined by the hyperelliptic equation y^2 = P(x), resp. y^2 + Q(x)*y = P(x). (The special fiber Xp of the minimal regular model X of C over ℤ.)

If p is omitted, determines the reduction type for all (odd) prime divisors of the discriminant.

This function was rewritten from an implementation of Liu's algorithm by Cohen and Liu (1994), `genus2reduction-0.3`, see `http://www.math.u-bordeaux.fr/~liu/G2R/`.

CAVEAT. The function interface may change: for the time being, it returns [N,FaN, T, V] where N is either the local conductor at p or the global conductor, FaN is its factorization, y^2 = T defines a minimal model over ℤ[1/2] and V describes the reduction type at the various considered p. Unfortunately, the program is not complete for p = 2, and we may return the odd part of the conductor only: this is the case if the factorization includes the (impossible) term 2-1; if the factorization contains another power of 2, then this is the exact local conductor at 2 and N is the global conductor.

      ? default(debuglevel, 1);
      ? genus2red(x^6 + 3*x^3 + 63, 3)
      (potential) stable reduction: [1, []]
      reduction at p: [III{9}] page 184, [3, 3], f = 10
      %1 = [59049, Mat([3, 10]), x^6 + 3*x^3 + 63, [3, [1, []],
             ["[III{9}] page 184", [3, 3]]]]
      ? [N, FaN, T, V] = genus2red(x^3-x^2-1, x^2-x);  \\ X1(13), global reduction
      p = 13
      (potential) stable reduction: [5, [Mod(0, 13), Mod(0, 13)]]
      reduction at p: [I{0}-II-0] page 159, [], f = 2
      ? N
      %3 = 169
      ? FaN
      %4 = Mat([13, 2])   \\ in particular, good reduction at 2 !
      ? T
      %5 = x^6 + 58*x^5 + 1401*x^4 + 18038*x^3 + 130546*x^2 + 503516*x + 808561
      ? V
      %6 = [[13, [5, [Mod(0, 13), Mod(0, 13)]], ["[I{0}-II-0] page 159", []]]]

We now first describe the format of the vector V = Vp in the case where p was specified (local reduction at p): it is a triple [p, stable, red]. The component stable = [type, vecj] contains information about the stable reduction after a field extension; depending on types, the stable reduction is

* 1: smooth (i.e. the curve has potentially good reduction). The Jacobian J(C) has potentially good reduction.

* 2: an elliptic curve E with an ordinary double point; vecj contains j mod p, the modular invariant of E. The (potential) semi-abelian reduction of J(C) is the extension of an elliptic curve (with modular invariant j mod p) by a torus.

* 3: a projective line with two ordinary double points. The Jacobian J(C) has potentially multiplicative reduction.

* 4: the union of two projective lines crossing transversally at three points. The Jacobian J(C) has potentially multiplicative reduction.

* 5: the union of two elliptic curves E1 and E2 intersecting transversally at one point; vecj contains their modular invariants j1 and j2, which may live in a quadratic extension of 𝔽p and need not be distinct. The Jacobian J(C) has potentially good reduction, isomorphic to the product of the reductions of E1 and E2.

* 6: the union of an elliptic curve E and a projective line which has an ordinary double point, and these two components intersect transversally at one point; vecj contains j mod p, the modular invariant of E. The (potential) semi-abelian reduction of J(C) is the extension of an elliptic curve (with modular invariant j mod p) by a torus.

* 7: as in type 6, but the two components are both singular. The Jacobian J(C) has potentially multiplicative reduction.

The component red = [NUtype, neron] contains two data concerning the reduction at p without any ramified field extension.

The NUtype is a `t_STR` describing the reduction at p of C, following Namikawa-Ueno, The complete classification of fibers in pencils of curves of genus two, Manuscripta Math., vol. 9, (1973), pages 143-186. The reduction symbol is followed by the corresponding page number or page range in this article.

The second datum neron is the group of connected components (over an algebraic closure of 𝔽p) of the Néron model of J(C), given as a finite abelian group (vector of elementary divisors).

If p = 2, the red component may be omitted altogether (and replaced by `[]`, in the case where the program could not compute it. When p was not specified, V is the vector of all Vp, for all considered p.

Notes about Namikawa-Ueno types.

* A lower index is denoted between braces: for instance, `[I{2}-II-5]` means `[I2-II-5]`.

* If K and K' are Kodaira symbols for singular fibers of elliptic curves, then `[K-K'-m]` and `[K'-K-m]` are the same.

We define a total ordering on Kodaira symbol by fixing `I` < `I*` < `II` < `II*`,.... If the reduction type is the same, we order by the number of components, e.g. `I`2 < `I`4, etc. Then we normalize our output so that K ≤ K'.

* `[K-K'--1]` is `[K-K'-α]` in the notation of Namikawa-Ueno.

* The figure `[2I0-m]` in Namikawa-Ueno, page 159, must be denoted by `[2I0-(m+1)]`.

The library syntax is `GEN genus2red(GEN PQ, GEN p = NULL)`.

#### hyperellcharpoly(X)

X being a non-singular hyperelliptic curve defined over a finite field, return the characteristic polynomial of the Frobenius automorphism. X can be given either by a squarefree polynomial P such that X: y^2 = P(x) or by a vector [P,Q] such that X: y^2 + Q(x) y = P(x) and Q^2+4 P is squarefree.

The library syntax is `GEN hyperellcharpoly(GEN X)`.

#### hyperellpadicfrobenius(Q, p, n)

Let X be the curve defined by y^2 = Q(x), where Q is a polynomial of degree d over ℚ and p ≥ d a prime such that X has good reduction at p return the matrix of the Frobenius endomorphism ϕ on the crystalline module Dp(X) = ℚp ⨂ H^1dR(X/ℚ) with respect to the basis of the given model (ω, x ω,...,xg-1 ω), where ω = dx/(2 y) is the invariant differential, where g is the genus of X (either d = 2 g+1 or d = 2 g+2). The characteristic polynomial of ϕ is the numerator of the zeta-function of the reduction of the curve X modulo p. The matrix is computed to absolute p-adic precision p^n.

The library syntax is `GEN hyperellpadicfrobenius(GEN Q, ulong p, long n)`.

#### hyperellratpoints(X, h, {flag = 0})

X being a non-singular hyperelliptic curve given by an integral model, return a vector containing the affine rational points on the curve of naive height less than h. If flag = 1, stop as soon as a point is found; return either an empty vector or a vector containing a single point.

X is given either by a squarefree polynomial P such that X: y^2 = P(x) or by a vector [P,Q] such that X: y^2+Q(x) y = P(x) and Q^2+4 P is squarefree.

The parameter h can be

* an integer H: find the points [n/d,y] whose abscissas x = n/d have naive height ( = max(|n|, d)) less than H;

* a vector [N,D] with D ≤ N: find the points [n/d,y] with |n| ≤ N, d ≤ D.

* a vector [N,[D1,D2]] with D1 < D2 ≤ N find the points [n/d,y] with |n| ≤ N and D1 ≤ d ≤ D2.

The library syntax is `GEN hyperellratpoints(GEN X, GEN h, long flag)`.