Properties

Label 2.11.af_r
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:  $1 - 5 x + 17 x^{2} - 55 x^{3} + 121 x^{4}$
Frobenius angles:  $\pm0.155832309789$, $\pm0.541099915094$
Angle rank:  $2$ (numerical)
Number field:  4.0.10525.1
Galois group:  $D_{4}$
Jacobians:  $9$
Isomorphism classes:  9

This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $79$ $15721$ $1727809$ $212626525$ $26115201904$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $7$ $131$ $1297$ $14523$ $162152$ $1776071$ $19488707$ $214364803$ $2358066097$ $25937454806$

Jacobians and polarizations

This isogeny class contains the Jacobians of 9 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{11}$.

Endomorphism algebra over $\F_{11}$
The endomorphism algebra of this simple isogeny class is 4.0.10525.1.

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_r$2$2.121.j_at