Abelian variety isogeny class 2.81.abf_pg over $\F_{3^{4}}$

• Label: 2.81.abf_pg
• Base field: $\F_{3^{4}}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{3^{4}}$
Dimension:	$2$
L-polynomial:	$( 1 - 9 x )^{2}( 1 - 13 x + 81 x^{2} )$
	$1 - 31 x + 396 x^{2} - 2511 x^{3} + 6561 x^{4}$
Frobenius angles:	$0$, $0$, $\pm0.243120792737$
Angle rank:	$1$ (numerical)
Jacobians:	$0$

This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.

Newton polygon

$p$-rank:	$1$
Slopes:	$[0, 1/2, 1/2, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4416$	$41952000$	$282165601536$	$1853017994112000$	$12157574551206148416$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$51$	$6393$	$530946$	$43046673$	$3486758331$	$282428611038$	$22876777221531$	$1853020017948513$	$150094633878696546$	$12157665450587960073$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3^{4}}$.

Endomorphism algebra over $\F_{3^{4}}$

The isogeny class factors as 1.81.as $\times$ 1.81.an and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.81.as : the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
• 1.81.an : \(\Q(\sqrt{-155}) \).

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
2.81.af_acu	$2$	(not in LMFDB)
2.81.f_acu	$2$	(not in LMFDB)
2.81.bf_pg	$2$	(not in LMFDB)
2.81.ae_bt	$3$	(not in LMFDB)