Properties

Label 2.11.ai_bi
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

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Invariants

Base field:  $\F_{11}$
Dimension:  $2$
L-polynomial:  $( 1 - 6 x + 11 x^{2} )( 1 - 2 x + 11 x^{2} )$
Frobenius angles:  $\pm0.140218899004$, $\pm0.402508885479$
Angle rank:  $2$ (numerical)
Jacobians:  8

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 8 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 60 15120 1826460 213857280 25884541500 3141328554000 380066253715740 45960389686149120 5559987016732558140 672744193373765898000

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 4 126 1372 14606 160724 1773198 19503404 214408606 2357977252 25937200926

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
The isogeny class factors as 1.11.ag $\times$ 1.11.ac and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
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.ae_k$2$2.121.e_ak
2.11.e_k$2$2.121.e_ak
2.11.i_bi$2$2.121.e_ak