Properties

Label 2.41.at_gj
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:  $1 - 19 x + 165 x^{2} - 779 x^{3} + 1681 x^{4}$
Frobenius angles:  $\pm0.0989395465808$, $\pm0.321602247780$
Angle rank:  $2$ (numerical)
Number field:  4.0.1518005.3
Galois group:  $D_{4}$
Jacobians:  8

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 8 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1049 2774605 4764483521 7987779813845 13421787842370304 22563041329388594125 37929203879206050681689 63759076936981080358956005 107178953321338722495950327441 180167788485924748792019195392000

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 23 1651 69131 2826771 115848678 4750009723 194754154163 7984930992771 327382002674471 13422659722105326

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
The endomorphism algebra of this simple isogeny class is 4.0.1518005.3.
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.t_gj$2$(not in LMFDB)