Properties

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

Learn more about

Invariants

Base field:  $\F_{11}$
Dimension:  $2$
L-polynomial:  $( 1 - 5 x + 11 x^{2} )^{2}$
Frobenius angles:  $\pm0.228229222880$, $\pm0.228229222880$
Angle rank:  $1$ (numerical)
Jacobians:  1

This isogeny class is not 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 1 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 49 14161 1882384 221265625 26171797729 3142195845376 379646013033049 45939009972515625 5559465921864288784 672739437300921924241

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 2 116 1412 15108 162502 1773686 19481842 214308868 2357756252 25937017556

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
The isogeny class factors as 1.11.af 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-19}) \)$)$
All geometric endomorphisms are defined over $\F_{11}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.11.a_ad$2$2.121.ag_jr
2.11.k_bv$2$2.121.ag_jr
2.11.f_o$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.11.a_ad$2$2.121.ag_jr
2.11.k_bv$2$2.121.ag_jr
2.11.f_o$3$(not in LMFDB)
2.11.a_d$4$(not in LMFDB)
2.11.af_o$6$(not in LMFDB)