Properties

Label 2.11.ak_bt
Base Field $\F_{11}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{11}$
Dimension:  $2$
L-polynomial:  $1 - 10 x + 45 x^{2} - 110 x^{3} + 121 x^{4}$
Frobenius angles:  $\pm0.0820279942768$, $\pm0.318205720493$
Angle rank:  $2$ (numerical)
Number field:  4.0.5696.1
Galois group:  $D_{4}$
Jacobians:  1

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 1 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 47 13489 1797092 214866281 25865323927 3133790594704 379659633349967 45953585808895625 5560281532649043332 672764077782820784929

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 2 112 1352 14676 160602 1768942 19482542 214376868 2358102152 25937967552

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
The endomorphism algebra of this simple isogeny class is 4.0.5696.1.
All geometric endomorphisms are defined over $\F_{11}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.11.k_bt$2$2.121.ak_cp