Properties

Label 2.11.af_p
Base field $\F_{11}$
Dimension $2$
$p$-rank $2$
Ordinary Yes
Supersingular No
Simple Yes
Geometrically simple Yes
Primitive Yes
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{11}$
Dimension:  $2$
L-polynomial:  $1 - 5 x + 15 x^{2} - 55 x^{3} + 121 x^{4}$
Frobenius angles:  $\pm0.123520620106$, $\pm0.554980822636$
Angle rank:  $2$ (numerical)
Number field:  4.0.755621.1
Galois group:  $D_{4}$
Jacobians:  7

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 7 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 77 15169 1688687 211592381 26082870352 3144076824889 379745616403307 45954823512723125 5560361340404247377 672754868614566056704

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 7 127 1267 14451 161952 1774747 19486957 214382643 2358135997 25937612502

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
The endomorphism algebra of this simple isogeny class is 4.0.755621.1.
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.f_p$2$2.121.f_adf