Properties

Label 2.41.as_gh
Base Field $\F_{41}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{41}$
Dimension:  $2$
L-polynomial:  $( 1 - 9 x + 41 x^{2} )^{2}$
Frobenius angles:  $\pm0.251940962052$, $\pm0.251940962052$
Angle rank:  $1$ (numerical)
Jacobians:  15

This isogeny class is not simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $2$
Slopes:  $[0, 0, 1, 1]$

Point counts

This isogeny class contains the Jacobians of 15 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1089 2832489 4802490000 8003936949129 13426077749128209 22563442409840640000 37928973949344795906369 63758940768104991107368329 107178915157433098275355290000 180167783335487620439204765476809

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 24 1684 69678 2832484 115885704 4750094158 194752973544 7984913939524 327381886101438 13422659338393204

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
The isogeny class factors as 1.41.aj 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-83}) \)$)$
All geometric endomorphisms are defined over $\F_{41}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.41.a_b$2$(not in LMFDB)
2.41.s_gh$2$(not in LMFDB)
2.41.j_bo$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.41.a_b$2$(not in LMFDB)
2.41.s_gh$2$(not in LMFDB)
2.41.j_bo$3$(not in LMFDB)
2.41.a_ab$4$(not in LMFDB)
2.41.aj_bo$6$(not in LMFDB)