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

• Label: 2.243.ace_bwn
• 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:	$( 1 - 31 x + 243 x^{2} )( 1 - 25 x + 243 x^{2} )$
	$1 - 56 x + 1261 x^{2} - 13608 x^{3} + 59049 x^{4}$
Frobenius angles:	$\pm0.0339262533067$, $\pm0.203835764481$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$46647$	$3450711825$	$205825251907728$	$12157634946045785625$	$717898201159914696700647$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$188$	$58436$	$14344316$	$3486775652$	$847288861388$	$205891131005414$	$50031544872384116$	$12157665451685114948$	$2954312706394075019588$	$717897987689350493333636$

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.az 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.az : \(\Q(\sqrt{-347}) \).

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.ag_ald	$2$	(not in LMFDB)
2.243.g_ald	$2$	(not in LMFDB)
2.243.ce_bwn	$2$	(not in LMFDB)