Properties

Label 2.13.aj_bp
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:  $1 - 9 x + 41 x^{2} - 117 x^{3} + 169 x^{4}$
Frobenius angles:  $\pm0.109149799241$, $\pm0.400911184348$
Angle rank:  $2$ (numerical)
Number field:  4.0.122157.1
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
$A(\F_{q^r})$ 85 28645 4886905 810796725 137471390800 23301803950765 3938925735789745 665496620891533125 112456859467929137365 19004959616264376217600

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 5 171 2225 28387 370250 4827579 62773205 815828803 10604636345 137858461686

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The endomorphism algebra of this simple isogeny class is 4.0.122157.1.
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.j_bp$2$2.169.b_adj