Abelian variety isogeny class 2.13.ab_z over $\F_{13}$

• Label: 2.13.ab_z
• Base field: $\F_{13}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{13}$
Dimension:	$2$
L-polynomial:	$1 - x + 25 x^{2} - 13 x^{3} + 169 x^{4}$
Frobenius angles:	$\pm0.427964010826$, $\pm0.527314542962$
Angle rank:	$2$ (numerical)
Number field:	4.0.63725.1
Galois group:	$D_{4}$
Jacobians:	$3$
Cyclic group of points:	yes

This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$181$	$37829$	$4906729$	$800802101$	$137637384016$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$13$	$219$	$2233$	$28035$	$370698$	$4831563$	$62755405$	$815700099$	$10604459089$	$137858503414$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 3 curves (of which all are hyperelliptic):

• $y^2=5 x^6+8 x^5+10 x^4+9 x^3+2 x^2+9 x+1$
• $y^2=7 x^6+7 x^5+4 x^4+7 x^3+6 x^2+2 x+8$
• $y^2=7 x^6+5 x^5+5 x^4+8 x^3+9 x^2+10 x+9$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{13}$.

Endomorphism algebra over $\F_{13}$

The endomorphism algebra of this simple isogeny class is 4.0.63725.1.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.13.b_z	$2$	2.169.bx_bkb