Properties

Label 2.13.ah_bg
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:  $( 1 - 6 x + 13 x^{2} )( 1 - x + 13 x^{2} )$
Frobenius angles:  $\pm0.187167041811$, $\pm0.455715642762$
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
$A(\F_{q^r})$ 104 31200 4954976 814320000 138011646344 23331990988800 3938757157931336 665396405968320000 112452136652458719584 19004887894305550380000

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 7 185 2254 28513 371707 4833830 62770519 815705953 10604190982 137857941425

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The isogeny class factors as 1.13.ag $\times$ 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.af_u$2$2.169.p_dk
2.13.f_u$2$2.169.p_dk
2.13.h_bg$2$2.169.p_dk
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.13.af_u$2$2.169.p_dk
2.13.f_u$2$2.169.p_dk
2.13.h_bg$2$2.169.p_dk
2.13.af_be$4$(not in LMFDB)
2.13.ad_w$4$(not in LMFDB)
2.13.d_w$4$(not in LMFDB)
2.13.f_be$4$(not in LMFDB)