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

• Label: 2.13.al_cd
• 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:	$1 - 11 x + 55 x^{2} - 143 x^{3} + 169 x^{4}$
Frobenius angles:	$\pm0.129998747777$, $\pm0.292104599859$
Angle rank:	$2$ (numerical)
Number field:	4.0.6725.1
Galois group:	$D_{4}$
Jacobians:	$1$
Isomorphism classes:	1
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$71$	$26909$	$4949339$	$824733941$	$138120708816$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$3$	$159$	$2253$	$28875$	$371998$	$4827063$	$62748045$	$815759379$	$10604757819$	$137859648214$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobian of 1 curve (which is hyperelliptic):

• $y^2=2 x^6+10 x^4+12 x+2$

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 4.0.6725.1.

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.l_cd	$2$	2.169.al_ij