Properties

Label 2.13.ah_ba
Base Field $\F_{13}$
Dimension $2$
Ordinary No
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{13}$
Dimension:  $2$
L-polynomial:  $( 1 - 7 x + 13 x^{2} )( 1 + 13 x^{2} )$
Frobenius angles:  $\pm0.0772104791556$, $\pm0.5$
Angle rank:  $1$ (numerical)
Jacobians:  2

This isogeny class is not simple.

Newton polygon

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

Point counts

This isogeny class contains the Jacobians of 2 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 98 28812 4677344 800743104 137700691898 23316859190016 3937502572968986 665386850963116800 112456662227757292256 19005143383543652756652

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 7 173 2128 28033 370867 4830698 62750527 815694241 10604617744 137859794693

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The isogeny class factors as 1.13.ah $\times$ 1.13.a and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{13}$
The base change of $A$ to $\F_{13^{2}}$ is 1.169.ax $\times$ 1.169.ba. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{13^{2}}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.13.h_ba$2$2.169.d_aka
2.13.c_ba$3$(not in LMFDB)
2.13.f_ba$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.13.h_ba$2$2.169.d_aka
2.13.c_ba$3$(not in LMFDB)
2.13.f_ba$3$(not in LMFDB)
2.13.af_ba$6$(not in LMFDB)
2.13.ac_ba$6$(not in LMFDB)