compute endomorphism orders for elliptic curves over finite fields (rank-2 case) #38493
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…ld (for the rank-2 case)
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I'll look at this tomorrow. I did implement a special case of Sutherland 's algorithm already for our CM testing algorithm last year. |
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Thank you! I think the implementation you wrote is the function I'm calling here. I just extended it to work for the supersingular case, too. |
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Apologies to @yyyyx4 for promising and then not delivering -- sheer forgetfulness on my part. I cannot promise to be able to review this in the next several days. Feel free to send me reminders. |
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Documentation preview for this PR (built with commit 0493778; changes) is ready! 🎉 |
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No worries, @JohnCremona! I do think it would be nice to get this into the next release, but it is by no means urgent. |
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For an elliptic curve E over a finite field with an endomorphism algebra of rank two over ℚ, the endomorphism ring is a superorder of the "Frobenius order" ℤ[π].
This patch adds a simple method to compute the exact imaginary-quadratic order containing ℤ[π] which is isomorphic to the endomorphism ring of the curve, and generalizes the algorithm to supersingular elliptic curves with a (rational) endomorphism algebra of rank 2.