Properties

Label 2.13.ah_bi
Base Field $\F_{13}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{13}$
Dimension:  $2$
L-polynomial:  $1 - 7 x + 34 x^{2} - 91 x^{3} + 169 x^{4}$
Frobenius angles:  $\pm0.219632524221$, $\pm0.436075769158$
Angle rank:  $2$ (numerical)
Number field:  4.0.304028.1
Galois group:  $D_{4}$
Jacobians:  4

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 4 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 106 32012 5048992 817970624 137907422626 23317174070528 3938240466496162 665384607166844672 112451105596046887072 19004836898524828256172

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 7 189 2296 28641 371427 4830762 62762287 815691489 10604093752 137857571509

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The endomorphism algebra of this simple isogeny class is 4.0.304028.1.
All geometric endomorphisms are defined over $\F_{13}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.13.h_bi$2$2.169.t_im