#9052: Hasse invariant for elliptic curves
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Reporter: voloch | Owner: cremona
Type: enhancement | Status: positive_review
Priority: minor | Milestone: sage-4.4.3
Component: elliptic curves | Keywords: Hasse invariant
Author: Felipe Voloch | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
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Comment(by cremona):
This is a coincidence, since just yesterday I was considering implementing
functions is_supersingular() and is_ordinary(). Now this can be done very
simply (since s.s. curves have invariant 0 and ordinary ones have nonzero
invariant).
However, I'm a little worried about the efficiency of the current
implementation for even modest p, since it involves raising a degree 3
polynomial to the power (p-1)/2 and then picking out one coefficient.
There are easier ways to test supersingularity for small p, since one can
precompute the s.s. j-invariants and check that. This would be a quicker
way of computing H when it is 0. One could check that the j-invariant has
degree at most 2 (else ordinary). And over the prime field GF(p), s.s.
curves have cardinality p+1, and another way to check ordinary-ness is to
take random points and multiply then by p+1. As a last resort one can
compute the cardinality.
I guess this is enough for a second ticket!
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/9052#comment:3>
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