Properties

Label 2.41.au_gy
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:  $1 - 20 x + 180 x^{2} - 820 x^{3} + 1681 x^{4}$
Frobenius angles:  $\pm0.149793866772$, $\pm0.266106536017$
Angle rank:  $2$ (numerical)
Number field:  4.0.194816.2
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
$A(\F_{q^r})$ 1022 2761444 4773640382 7997152869776 13425683554533502 22563916483823527204 37929240932572054753982 63759026080220113287680000 107178932568501927486017183582 180167784484024282754831215495204

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 22 1642 69262 2830086 115882302 4750193962 194754344422 7984924623678 327381939284182 13422659423960202

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
The endomorphism algebra of this simple isogeny class is 4.0.194816.2.
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.u_gy$2$(not in LMFDB)