Weierstrass
-function for elliptic curves¶
The Weierstrass function associated to an elliptic curve over a
field
is a Laurent series of the form
If the field is contained in , then this is the series
expansion of the map from
to
whose kernel
is the period lattice of
.
Over other fields, like finite fields, this still makes sense as a
formal power series with coefficients in - at least its first
coefficients where
is the characteristic of
. It can be defined
via the formal group as
in the variable
for a
constant
such that the constant term
in
is zero.
EXAMPLES:
sage: E = EllipticCurve([0,1])
sage: E.weierstrass_p()
z^-2 - 1/7*z^4 + 1/637*z^10 - 1/84721*z^16 + O(z^20)
REFERENCES:
- [BMSS] Boston, Morain, Salvy, Schost, “Fast Algorithms for Isogenies.”
AUTHORS:
- Dan Shumov 04/09: original implementation
- Chris Wuthrich 11/09: major restructuring
- Jeroen Demeyer (2014-03-06): code clean up, fix characteristic bound for quadratic algorithm (see trac ticket #15855)
-
sage.schemes.elliptic_curves.ell_wp.
compute_wp_fast
(k, A, B, m)¶ Computes the Weierstrass function of an elliptic curve defined by short Weierstrass model:
. It does this with as fast as polynomial of degree
can be multiplied together in the base ring, i.e.
in the notation of [BMSS].
Let
be the characteristic of the underlying field: Then we must have either
, or
.
INPUT:
k
- the base field of the curveA
- andB
- as the coeffients of the short Weierstrass model, and
m
- the precision to which the function is computed to.
OUTPUT:
the Weierstrass
function as a Laurent series to precision
.
ALGORITHM:
This function uses the algorithm described in section 3.3 of [BMSS].
EXAMPLES:
sage: from sage.schemes.elliptic_curves.ell_wp import compute_wp_fast sage: compute_wp_fast(QQ, 1, 8, 7) z^-2 - 1/5*z^2 - 8/7*z^4 + 1/75*z^6 + O(z^7) sage: k = GF(37) sage: compute_wp_fast(k, k(1), k(8), 5) z^-2 + 22*z^2 + 20*z^4 + O(z^5)
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sage.schemes.elliptic_curves.ell_wp.
compute_wp_pari
(E, prec)¶ Computes the Weierstrass
-function with the
ellwp
function from PARI.EXAMPLES:
sage: E = EllipticCurve([0,1]) sage: from sage.schemes.elliptic_curves.ell_wp import compute_wp_pari sage: compute_wp_pari(E, prec=20) z^-2 - 1/7*z^4 + 1/637*z^10 - 1/84721*z^16 + O(z^20) sage: compute_wp_pari(E, prec=30) z^-2 - 1/7*z^4 + 1/637*z^10 - 1/84721*z^16 + 3/38548055*z^22 - 4/8364927935*z^28 + O(z^30)
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sage.schemes.elliptic_curves.ell_wp.
compute_wp_quadratic
(k, A, B, prec)¶ Computes the truncated Weierstrass function of an elliptic curve defined by short Weierstrass model:
. Uses an algorithm that is of complexity
.
Let p be the characteristic of the underlying field. Then we must have either p = 0, or p > prec + 2.
INPUT:
k
- the field of definition of the curveA
- andB
- the coefficients of the elliptic curveprec
- the precision to which we compute the series.
OUTPUT: A Laurent series aproximating the Weierstrass
-function to precision
prec
.ALGORITHM: This function uses the algorithm described in section 3.2 of [BMSS].
REFERENCES: [BMSS] Boston, Morain, Salvy, Schost, “Fast Algorithms for Isogenies.”
EXAMPLES:
sage: E = EllipticCurve([7,0]) sage: E.weierstrass_p(prec=10, algorithm='quadratic') z^-2 - 7/5*z^2 + 49/75*z^6 + O(z^10) sage: E = EllipticCurve(GF(103),[1,2]) sage: E.weierstrass_p(algorithm='quadratic') z^-2 + 41*z^2 + 88*z^4 + 11*z^6 + 57*z^8 + 55*z^10 + 73*z^12 + 11*z^14 + 17*z^16 + 50*z^18 + O(z^20) sage: from sage.schemes.elliptic_curves.ell_wp import compute_wp_quadratic sage: compute_wp_quadratic(E.base_ring(), E.a4(), E.a6(), prec=10) z^-2 + 41*z^2 + 88*z^4 + 11*z^6 + 57*z^8 + O(z^10)
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sage.schemes.elliptic_curves.ell_wp.
solve_linear_differential_system
(a, b, c, alpha)¶ Solves a system of linear differential equations:
and
where
,
, and
are power series in one variable and
is a constant in the coefficient ring.
ALGORITHM:
due to Brent and Kung ‘78.
EXAMPLES:
sage: from sage.schemes.elliptic_curves.ell_wp import solve_linear_differential_system sage: k = GF(17) sage: R.<x> = PowerSeriesRing(k) sage: a = 1+x+O(x^7); b = x+O(x^7); c = 1+x^3+O(x^7); alpha = k(3) sage: f = solve_linear_differential_system(a,b,c,alpha) sage: f 3 + x + 15*x^2 + x^3 + 10*x^5 + 3*x^6 + 13*x^7 + O(x^8) sage: a*f.derivative()+b*f - c O(x^7) sage: f(0) == alpha True
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sage.schemes.elliptic_curves.ell_wp.
weierstrass_p
(E, prec=20, algorithm=None)¶ Computes the Weierstrass
-function on an elliptic curve.
INPUT:
E
– an elliptic curveprec
– precisionalgorithm
– string (default:None
) an algorithm identifier indicating thepari
,fast
orquadratic
algorithm. If the algorithm isNone
, then this function determines the best algorithm to use.
OUTPUT:
a Laurent series in one variable
with coefficients in the base field
of
.
EXAMPLES:
sage: E = EllipticCurve('11a1') sage: E.weierstrass_p(prec=10) z^-2 + 31/15*z^2 + 2501/756*z^4 + 961/675*z^6 + 77531/41580*z^8 + O(z^10) sage: E.weierstrass_p(prec=8) z^-2 + 31/15*z^2 + 2501/756*z^4 + 961/675*z^6 + O(z^8) sage: Esh = E.short_weierstrass_model() sage: Esh.weierstrass_p(prec=8) z^-2 + 13392/5*z^2 + 1080432/7*z^4 + 59781888/25*z^6 + O(z^8) sage: E.weierstrass_p(prec=8, algorithm='pari') z^-2 + 31/15*z^2 + 2501/756*z^4 + 961/675*z^6 + O(z^8) sage: E.weierstrass_p(prec=8, algorithm='quadratic') z^-2 + 31/15*z^2 + 2501/756*z^4 + 961/675*z^6 + O(z^8) sage: k = GF(11) sage: E = EllipticCurve(k, [1,1]) sage: E.weierstrass_p(prec=6, algorithm='fast') z^-2 + 2*z^2 + 3*z^4 + O(z^6) sage: E.weierstrass_p(prec=7, algorithm='fast') Traceback (most recent call last): ... ValueError: for computing the Weierstrass p-function via the fast algorithm, the characteristic (11) of the underlying field must be greater than prec + 4 = 11 sage: E.weierstrass_p(prec=8) z^-2 + 2*z^2 + 3*z^4 + 5*z^6 + O(z^8) sage: E.weierstrass_p(prec=8, algorithm='quadratic') z^-2 + 2*z^2 + 3*z^4 + 5*z^6 + O(z^8) sage: E.weierstrass_p(prec=8, algorithm='pari') z^-2 + 2*z^2 + 3*z^4 + 5*z^6 + O(z^8) sage: E.weierstrass_p(prec=9) Traceback (most recent call last): ... NotImplementedError: currently no algorithms for computing the Weierstrass p-function for that characteristic / precision pair is implemented. Lower the precision below char(k) - 2 sage: E.weierstrass_p(prec=9, algorithm="quadratic") Traceback (most recent call last): ... ValueError: for computing the Weierstrass p-function via the quadratic algorithm, the characteristic (11) of the underlying field must be greater than prec + 2 = 11 sage: E.weierstrass_p(prec=9, algorithm='pari') Traceback (most recent call last): ... ValueError: for computing the Weierstrass p-function via pari, the characteristic (11) of the underlying field must be greater than prec + 2 = 11