Abelian variety isogeny class 1.8.f over $\F_{2^{3}}$

• Label: 1.8.f
• Base field: $\F_{2^{3}}$
• Dimension: $1$
• $p$-rank: $1$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: no
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{2^{3}}$
Dimension:	$1$
L-polynomial:	$1 + 5 x + 8 x^{2}$
Frobenius angles:	$\pm0.845080184244$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-7}) \)
Galois group:	$C_2$
Jacobians:	1

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 1 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$14$	$56$	$518$	$4144$	$32494$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$14$	$56$	$518$	$4144$	$32494$	$263144$	$2094358$	$16783200$	$134210174$	$1073731736$

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{3}}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-7}) \).

All geometric endomorphisms are defined over $\F_{2^{3}}$.

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{3}}$.

Subfield	Primitive Model
$\F_{2}$	1.2.ab

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
1.8.af	$2$	1.64.aj