Properties

Label 2.41.at_gk
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:  $( 1 - 12 x + 41 x^{2} )( 1 - 7 x + 41 x^{2} )$
Frobenius angles:  $\pm0.113551764296$, $\pm0.315918729109$
Angle rank:  $2$ (numerical)
Jacobians:  10

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 10 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1050 2778300 4768444800 7989990724800 13422569265656250 22563223610314060800 37929229263299849875050 63759078059195325925036800 107178953226187440605312188800 180167788518282043328992436437500

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 23 1653 69188 2827553 115855423 4750048098 194754284503 7984931133313 327382002383828 13422659724515973

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
The isogeny class factors as 1.41.am $\times$ 1.41.ah and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{41}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.41.af_ac$2$(not in LMFDB)
2.41.f_ac$2$(not in LMFDB)
2.41.t_gk$2$(not in LMFDB)