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

• Label: 2.243.acb_bsy
• 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:	$( 1 - 31 x + 243 x^{2} )( 1 - 22 x + 243 x^{2} )$
	$1 - 53 x + 1168 x^{2} - 12879 x^{3} + 59049 x^{4}$
Frobenius angles:	$\pm0.0339262533067$, $\pm0.250654960462$
Angle rank:	$2$ (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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$47286$	$3458970900$	$205865265295368$	$12157702298182980000$	$717897735722275794043146$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$191$	$58577$	$14347106$	$3486794969$	$847288312061$	$205891108713314$	$50031544449630935$	$12157665447495817841$	$2954312706412989486998$	$717897987691067647237457$

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.abf $\times$ 1.243.aw and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.243.abf : \(\Q(\sqrt{-11}) \).
• 1.243.aw : \(\Q(\sqrt{-122}) \).

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.aj_aho	$2$	(not in LMFDB)
2.243.j_aho	$2$	(not in LMFDB)
2.243.cb_bsy	$2$	(not in LMFDB)