Ticket #9646 (closed defect: fixed)
Incorrect calculation of elliptic curve formal group law
|Reported by:||hlaw||Owned by:||cremona|
|Component:||elliptic curves||Keywords:||elliptic curve formal group law|
|Report Upstream:||N/A||Reviewers:||David Loeffler|
|Authors:||Chris Wuthrich||Merged in:||sage-4.6.2.alpha3|
Description (last modified by hlaw) (diff)
If F(t1, t2) is a formal group law, then F(t1, t2) = t1 + t2 (mod t1*t2). So in particular, the coefficients of t1^i and t2^i are zero for all i > 1. However the formal group law of an elliptic curve as returned by Sage includes (at least) the terms -a1^2*t1^3 and -a1^2*t2^2, as the following example shows:
sage: P.<a1, a2, a3, a4, a6> = PolynomialRing(ZZ, 5) sage: E = EllipticCurve(list(P.gens())) sage: F = E.formal().group_law(prec = 4) sage: t2 = F.parent().gen() sage: t1 = F.parent().base_ring().gen() sage: F(t1, 0) t1 - a1^2*t1^3 # should be t1 sage: F(0, t2) t2 - a1^2*t2^3
Note also that the coefficient of t1^2*t2 + t1*t2^2 returned by sage is -a1^2 - a2, whereas, according to Silverman AEC IV.1, it should be simply -a2.
This was obtained in Sage 4.4.4 on MacOS X 10.5.8 (32 bit).
- Status changed from needs_review to positive_review
- Reviewers set to David Loeffler
- Authors set to Chris Wuthrich