Abelian variety isogeny class 1.307.at over $\F_{307}$

• Label: 1.307.at
• Base field: $\F_{307}$
• 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:	$\F_{307}$
Dimension:	$1$
L-polynomial:	$1 - 19 x + 307 x^{2}$
Frobenius angles:	$\pm0.317593919300$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}) \)
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$289$	$94503$	$28945084$	$8882998491$	$2727041417119$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$289$	$94503$	$28945084$	$8882998491$	$2727041417119$	$837201936379536$	$257021010683307901$	$78905450519909984883$	$24223973309196979359748$	$7436759805841859050369143$

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 $\F_{307}$.

Endomorphism algebra over $\F_{307}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}) \).

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
1.307.t	$2$	(not in LMFDB)
1.307.aq	$3$	(not in LMFDB)
1.307.bj	$3$	(not in LMFDB)