Properties

Label 2.13.ai_bh
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:  $( 1 - 7 x + 13 x^{2} )( 1 - x + 13 x^{2} )$
Frobenius angles:  $\pm0.0772104791556$, $\pm0.455715642762$
Angle rank:  $2$ (numerical)
Jacobians:  6

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 6 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 91 28665 4758208 802190025 137411044771 23309890007040 3938324793124267 665412853037812425 112454588112991214272 19005059294898370518825

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 6 172 2166 28084 370086 4829254 62763630 815726116 10604422158 137859184732

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The isogeny class factors as 1.13.ah $\times$ 1.13.ab 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 $\F_{13}$.

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.ag_t$2$2.169.c_ajd
2.13.g_t$2$2.169.c_ajd
2.13.i_bh$2$2.169.c_ajd
2.13.b_y$3$(not in LMFDB)
2.13.e_v$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.13.ag_t$2$2.169.c_ajd
2.13.g_t$2$2.169.c_ajd
2.13.i_bh$2$2.169.c_ajd
2.13.b_y$3$(not in LMFDB)
2.13.e_v$3$(not in LMFDB)
2.13.ag_bf$6$(not in LMFDB)
2.13.ae_v$6$(not in LMFDB)
2.13.ad_bc$6$(not in LMFDB)
2.13.ab_y$6$(not in LMFDB)
2.13.d_bc$6$(not in LMFDB)
2.13.g_bf$6$(not in LMFDB)