Abelian variety isogeny class 1.512.af over $\F_{2^{9}}$

• Label: 1.512.af
• Base field: $\F_{2^{9}}$
• 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:	$\F_{2^{9}}$
Dimension:	$1$
L-polynomial:	$1 - 5 x + 512 x^{2}$
Frobenius angles:	$\pm0.464759447268$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-7}) \)
Galois group:	$C_2$

This isogeny class is simple and geometrically simple, not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$508$	$263144$	$134225284$	$68719003024$	$35184365852108$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$508$	$263144$	$134225284$	$68719003024$	$35184365852108$	$18014398720839416$	$9223372041104766164$	$4722366482782680160800$	$2417851639226647529086108$	$1237940039285411746904576264$

Jacobians and polarizations

This isogeny class contains a Jacobian, and hence is principally polarizable.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{9}}$.

Endomorphism algebra over $\F_{2^{9}}$

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

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{9}}$.

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

Twists

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

Twist	Extension degree	Common base change
1.512.f	$2$	(not in LMFDB)