Properties

Label 2.41.au_gu
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:  $1 - 20 x + 176 x^{2} - 820 x^{3} + 1681 x^{4}$
Frobenius angles:  $\pm0.0753103869890$, $\pm0.299287056274$
Angle rank:  $2$ (numerical)
Number field:  4.0.555264.3
Galois group:  $D_{4}$
Jacobians:  4

This isogeny class is simple and geometrically 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 4 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1018 2746564 4756935850 7987106988304 13421744275906378 22562867174484482500 37929077557742789172058 63759030482297998070267904 107178944014979710786024358650 180167788286418766789183391503684

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 22 1634 69022 2826534 115848302 4749973058 194753505542 7984925174974 327381974247862 13422659707241954

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
The endomorphism algebra of this simple isogeny class is 4.0.555264.3.
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.u_gu$2$(not in LMFDB)