Properties

Label 2.41.as_fx
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:  $1 - 18 x + 153 x^{2} - 738 x^{3} + 1681 x^{4}$
Frobenius angles:  $\pm0.101373436493$, $\pm0.349335512379$
Angle rank:  $2$ (numerical)
Number field:  4.0.3342400.2
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
$A(\F_{q^r})$ 1079 2795689 4764971900 7985161545049 13420968266033159 22563083485887552400 37929303017347036212359 63759104377295843693028969 107178953023456159915347203900 180167786128645038389263878156409

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 24 1664 69138 2825844 115841604 4750018598 194754663204 7984934429284 327382001764578 13422659546485904

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
The endomorphism algebra of this simple isogeny class is 4.0.3342400.2.
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.s_fx$2$(not in LMFDB)