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

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

Invariants

Base field:	$\F_{3^{5}}$
Dimension:	$2$
L-polynomial:	$( 1 - 29 x + 243 x^{2} )^{2}$
	$1 - 58 x + 1327 x^{2} - 14094 x^{3} + 59049 x^{4}$
Frobenius angles:	$\pm0.119654564389$, $\pm0.119654564389$
Angle rank:	$1$ (numerical)

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$46225$	$3445103025$	$205797960835600$	$12157610186613425625$	$717898935608159351880625$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$186$	$58340$	$14342412$	$3486768548$	$847289728206$	$205891168391270$	$50031545879742522$	$12157665472878391748$	$2954312706761935620276$	$717897987694615929001700$

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 isogeny class factors as 1.243.abd^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-131}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.243.a_anr	$2$	(not in LMFDB)
2.243.cg_bzb	$2$	(not in LMFDB)
2.243.bd_xa	$3$	(not in LMFDB)