Abelian variety isogeny class 1.343.ax over $\F_{7^{3}}$

• Label: 1.343.ax
• Base field: $\F_{7^{3}}$
• 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:	$\F_{7^{3}}$
Dimension:	$1$
L-polynomial:	$1 - 23 x + 343 x^{2}$
Frobenius angles:	$\pm0.286750387523$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-843}) \)
Galois group:	$C_2$
Jacobians:	$6$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$321$	$117807$	$40365108$	$13841497851$	$4747562410371$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$321$	$117807$	$40365108$	$13841497851$	$4747562410371$	$1628413546367664$	$558545862588953469$	$191581231363875987603$	$65712362363662955698044$	$22539340290700942442100807$

Jacobians and polarizations

This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{7^{3}}$.

Endomorphism algebra over $\F_{7^{3}}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-843}) \).

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
1.343.x	$2$	(not in LMFDB)