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

• Label: 2.243.acc_bus
• 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:	$( 1 - 28 x + 243 x^{2} )( 1 - 26 x + 243 x^{2} )$
	$1 - 54 x + 1214 x^{2} - 13122 x^{3} + 59049 x^{4}$
Frobenius angles:	$\pm0.144947286894$, $\pm0.186073871252$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$47088$	$3458142720$	$205888834147248$	$12158053519642214400$	$717900522398544136272048$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$190$	$58562$	$14348746$	$3486895694$	$847291600990$	$205891185219794$	$50031545805960202$	$12157665465422255966$	$2954312706556771665598$	$717897987690585480782882$

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.abc $\times$ 1.243.aba 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.abc : \(\Q(\sqrt{-47}) \).
• 1.243.aba : \(\Q(\sqrt{-74}) \).

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.ac_aji	$2$	(not in LMFDB)
2.243.c_aji	$2$	(not in LMFDB)
2.243.cc_bus	$2$	(not in LMFDB)