Abelian variety isogeny class 2.243.acg_byz over $\F_{3^{5}}$

• Label: 2.243.acg_byz
• Base field: $\F_{3^{5}}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{3^{5}}$
Dimension:	$2$
L-polynomial:	$1 - 58 x + 1325 x^{2} - 14094 x^{3} + 59049 x^{4}$
Frobenius angles:	$\pm0.0705529335341$, $\pm0.154284978955$
Angle rank:	$2$ (numerical)
Number field:	4.0.634432.1
Galois group:	$D_{4}$

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:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$46223$	$3444861521$	$205792962124964$	$12157553334846324841$	$717898466293546771418023$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$186$	$58336$	$14342064$	$3486752244$	$847289174306$	$205891153167646$	$50031545523687830$	$12157665465627182948$	$2954312706632844946848$	$717897987692644869473296$

Jacobians and polarizations

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3^{5}}$.

Endomorphism algebra over $\F_{3^{5}}$

The endomorphism algebra of this simple isogeny class is 4.0.634432.1.

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
2.243.cg_byz	$2$	(not in LMFDB)