Abelian variety isogeny class 1.313.aj over $\F_{313}$

• Label: 1.313.aj
• Base field: $\F_{313}$
• 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:	$\F_{313}$
Dimension:	$1$
L-polynomial:	$1 - 9 x + 313 x^{2}$
Frobenius angles:	$\pm0.418136824861$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-1171}) \)
Galois group:	$C_2$
Jacobians:	$7$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$305$	$98515$	$30672020$	$9597823875$	$3004147186025$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$305$	$98515$	$30672020$	$9597823875$	$3004147186025$	$940299112203520$	$294313622644389905$	$92120163565964119875$	$28833611193086972752820$	$9024920303509385765707075$

Jacobians and polarizations

This isogeny class contains the Jacobians of 7 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{313}$.

Endomorphism algebra over $\F_{313}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-1171}) \).

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
1.313.j	$2$	(not in LMFDB)