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

• Label: 2.243.acc_bud
• 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:	$( 1 - 31 x + 243 x^{2} )( 1 - 23 x + 243 x^{2} )$
	$1 - 54 x + 1199 x^{2} - 13122 x^{3} + 59049 x^{4}$
Frobenius angles:	$\pm0.0339262533067$, $\pm0.235899952404$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$47073$	$3456335025$	$205853935339728$	$12157695865264145625$	$717897962480854676711073$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$190$	$58532$	$14346316$	$3486793124$	$847288579690$	$205891116605414$	$50031544565761102$	$12157665447928180676$	$2954312706387741577588$	$717897987690306450144932$

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.ax 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.ax : \(\Q(\sqrt{-443}) \).

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.ai_ait	$2$	(not in LMFDB)
2.243.i_ait	$2$	(not in LMFDB)
2.243.cc_bud	$2$	(not in LMFDB)