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j-invariant of Fp2

asked 2020-04-04 19:53:45 +0200

niranjan gravatar image


Can you please give a pointer to how to calculate set of supersingular j-invariants of quadratic extension of prime fields? Specifically for Fp2 = Fp(i) with elements of the form 'u+iv' where u,v belongs to Fp.

Thanks Niranjan

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answered 2020-04-05 17:48:00 +0200

Sébastien gravatar image

updated 2020-04-06 09:40:56 +0200

The function supersingular_j returns one such supersingular j-invariant :

sage: supersingular_j(GF(19^2, 'i'))
sage: supersingular_j(GF(15073^2, 'i'))
4443*i + 13964

To get all of them you use the method supersingular_points:

sage: S = SupersingularModule(19)
sage: S
Module of supersingular points on X_0(1)/F_19 over Integer Ring
sage: L,d = S.supersingular_points()
sage: L
[18, 7]
sage: S = SupersingularModule(431)
sage: L,d = S.supersingular_points()
sage: len(L)
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Thanks a lot, @Sebastien.

niranjan gravatar imageniranjan ( 2020-04-06 07:43:06 +0200 )edit

Is there a method to get all or a list of j-variants? For example, GF(431^2, 'i') has 37 j-variants. Above function just returns one (e.g. 4). How can we get a subset of 37? Thanks.

niranjan gravatar imageniranjan ( 2020-04-06 09:25:34 +0200 )edit

The documentation I linked to contains the answer. I updated the answer to return the list. Note that I do not know the math involved, so please check that it does what you want, I only know how to search/read the documentation.

Sébastien gravatar imageSébastien ( 2020-04-06 09:43:07 +0200 )edit

@Sebastien: thank you so much! This code helps. And, my apologies for not looking into the documentation carefully.

niranjan gravatar imageniranjan ( 2020-04-06 11:32:13 +0200 )edit

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Asked: 2020-04-04 19:53:45 +0200

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Last updated: Apr 06 '20