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

• Label: 2.13.aj_bp
• 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:	$1 - 9 x + 41 x^{2} - 117 x^{3} + 169 x^{4}$
Frobenius angles:	$\pm0.109149799241$, $\pm0.400911184348$
Angle rank:	$2$ (numerical)
Number field:	4.0.122157.1
Galois group:	$D_{4}$
Jacobians:	$2$
Isomorphism classes:	2

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$85$	$28645$	$4886905$	$810796725$	$137471390800$

Point counts of the curve

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

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which all are hyperelliptic):

• $y^2=2 x^6+x^5+10 x^4+4 x^3+10 x+11$
• $y^2=11 x^6+7 x^5+12 x^4+8 x^3+3 x^2+6 x+8$

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 4.0.122157.1.

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.j_bp	$2$	2.169.b_adj