Properties

Label 2.11.ak_bt
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

Related objects

Downloads

Learn more

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:  $1$
Isomorphism classes:  1

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $47$ $13489$ $1797092$ $214866281$ $25865323927$

Point counts of the curve

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

Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is 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 4.0.5696.1.

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.k_bt$2$2.121.ak_cp