Properties

Label 2.13.ai_bj
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:  $1 - 8 x + 35 x^{2} - 104 x^{3} + 169 x^{4}$
Frobenius angles:  $\pm0.126882739163$, $\pm0.439864156467$
Angle rank:  $2$ (numerical)
Number field:  4.0.308112.1
Galois group:  $D_{4}$
Jacobians:  2

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 2 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 93 29481 4864644 808988121 137677549773 23322601646352 3939273752827989 665460387626023593 112455211642756494948 19004999707306338623241

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 6 176 2214 28324 370806 4831886 62778750 815784388 10604480958 137858752496

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The endomorphism algebra of this simple isogeny class is 4.0.308112.1.
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.i_bj$2$2.169.g_adx