Properties

Label 2.11.ag_w
Base field $\F_{11}$
Dimension $2$
$p$-rank $1$
Ordinary No
Supersingular No
Simple No
Geometrically simple No
Primitive Yes
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{11}$
Dimension:  $2$
L-polynomial:  $( 1 - 6 x + 11 x^{2} )( 1 + 11 x^{2} )$
Frobenius angles:  $\pm0.140218899004$, $\pm0.5$
Angle rank:  $1$ (numerical)
Jacobians:  16

This isogeny class is not simple.

Newton polygon

$p$-rank:  $1$
Slopes:  $[0, 1/2, 1/2, 1]$

Point counts

This isogeny class contains the Jacobians of 16 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 72 15552 1750248 211507200 26014085352 3147295953600 379921632491592 45949275913420800 5560073041789601928 672760876491684395712

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 6 130 1314 14446 161526 1776562 19495986 214356766 2358013734 25937844130

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
The isogeny class factors as 1.11.ag $\times$ 1.11.a and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{11}$
The base change of $A$ to $\F_{11^{2}}$ is 1.121.ao $\times$ 1.121.w. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{11^{2}}$.

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.g_w$2$2.121.i_aco