Properties

Label 2.41.at_gl
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:  $1 - 19 x + 167 x^{2} - 779 x^{3} + 1681 x^{4}$
Frobenius angles:  $\pm0.127582089543$, $\pm0.309683357545$
Angle rank:  $2$ (numerical)
Number field:  4.0.1233477.1
Galois group:  $D_{4}$
Jacobians:  6

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 6 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1051 2781997 4772407075 7992190531525 13423328697807856 22563384488596700725 37929240919197854613451 63759072754788643729282725 107178950848637129350740973675 180167787937613187857793841837312

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 23 1655 69245 2828331 115861978 4750081967 194754344353 7984930469011 327381995121515 13422659681255630

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
The endomorphism algebra of this simple isogeny class is 4.0.1233477.1.
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_gl$2$(not in LMFDB)