Abelian variety isogeny class 1.13.b over $\F_{13}$

• Label: 1.13.b
• Base field: $\F_{13}$
• Dimension: $1$
• $p$-rank: $1$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{13}$
Dimension:	$1$
L-polynomial:	$1 + x + 13 x^{2}$
Frobenius angles:	$\pm0.544284357238$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-51}) \)
Galois group:	$C_2$
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:	$1$
Slopes:	$[0, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$15$	$195$	$2160$	$28275$	$372075$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$15$	$195$	$2160$	$28275$	$372075$	$4829760$	$62735415$	$815705475$	$10604694960$	$137858624475$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which 0 are hyperelliptic):

• $y^2=x^3+2 x+2$
• $y^2=x^3+4 x+4$

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 \(\Q(\sqrt{-51}) \).

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
1.13.ab	$2$	1.169.z