Properties

Label 2.41.at_gi
Base Field $\F_{41}$
Dimension $2$
Ordinary No
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{41}$
Dimension:  $2$
L-polynomial:  $1 - 19 x + 164 x^{2} - 779 x^{3} + 1681 x^{4}$
Frobenius angles:  $\pm0.0831029995597$, $\pm0.326848656985$
Angle rank:  $2$ (numerical)
Number field:  4.0.357192.2
Galois group:  $D_{4}$
Jacobians:  6

This isogeny class is simple and geometrically simple.

Newton polygon

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

Point counts

This isogeny class contains the Jacobians of 6 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1048 2770912 4760523232 7985557794688 13420984426801048 22562837588624171008 37929164611481170118488 63759069174851069175111168 107178950938488724582675057888 180167787706485985312147331307232

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 23 1649 69074 2825985 115841743 4749966830 194753952535 7984930020673 327381995395970 13422659664036449

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
The endomorphism algebra of this simple isogeny class is 4.0.357192.2.
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.t_gi$2$(not in LMFDB)