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

• Label: 2.243.ace_bwr
• 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:	$1 - 56 x + 1265 x^{2} - 13608 x^{3} + 59049 x^{4}$
Frobenius angles:	$\pm0.0783982229649$, $\pm0.190397219869$
Angle rank:	$2$ (numerical)
Number field:	4.0.445225.1
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$46651$	$3451194329$	$205834903662784$	$12157739442976229641$	$717899010627359245105091$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$188$	$58444$	$14344988$	$3486805620$	$847289816748$	$205891155303238$	$50031545391647700$	$12157665461262443684$	$2954312706548675193764$	$717897987691553646133564$

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 4.0.445225.1.

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.ce_bwr	$2$	(not in LMFDB)