Properties

Label 2.11.af_s
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:  $1 - 5 x + 18 x^{2} - 55 x^{3} + 121 x^{4}$
Frobenius angles:  $\pm0.170748385587$, $\pm0.533728959213$
Angle rank:  $2$ (numerical)
Number field:  4.0.8405.1
Galois group:  $D_{4}$
Jacobians:  10

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 10 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 80 16000 1747520 213056000 26119222000 3146934016000 379774285897520 45948141875456000 5560077704924386880 672749013766810000000

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 7 133 1312 14553 162177 1776358 19488427 214351473 2358015712 25937386773

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
The endomorphism algebra of this simple isogeny class is 4.0.8405.1.
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_s$2$2.121.l_q