Abelian variety isogeny class 1.7.e over $\F_{7}$

• Label: 1.7.e
• Base field: $\F_{7}$
• 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:	$\F_{7}$
Dimension:	$1$
L-polynomial:	$1 + 4 x + 7 x^{2}$
Frobenius angles:	$\pm0.772814474171$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}) \)
Galois group:	$C_2$
Jacobians:	2

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$1$
Slopes:	$[0, 1]$

Point counts

This isogeny class contains the Jacobians of 2 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$12$	$48$	$324$	$2496$	$16572$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$12$	$48$	$324$	$2496$	$16572$	$117936$	$824052$	$5760768$	$40366188$	$282453168$

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7}$

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

All geometric endomorphisms are defined over $\F_{7}$.

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
1.7.ae	$2$	1.49.ac
1.7.af	$3$	1.343.au
1.7.b	$3$	1.343.au