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

• Label: 2.81.abk_ss
• Base field: $\F_{3^{4}}$
• Dimension: $2$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: no
• Geometrically simple: no
• Primitive: no
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{3^{4}}$
Dimension:	$2$
L-polynomial:	$( 1 - 9 x )^{4}$
	$1 - 36 x + 486 x^{2} - 2916 x^{3} + 6561 x^{4}$
Frobenius angles:	$0$, $0$, $0$, $0$
Angle rank:	$0$ (numerical)
Jacobians:	$1$

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

Newton polygon

This isogeny class is supersingular.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4096$	$40960000$	$280883040256$	$1851890728960000$	$12156841915449020416$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$46$	$6238$	$528526$	$43020478$	$3486548206$	$282427410718$	$22876773323086$	$1853020016664958$	$150094633747317166$	$12157665445109791198$

Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

• $y^2=ax^5+2a$

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^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{3^{4}}$.

Subfield	Primitive Model
$\F_{3}$	2.3.a_ag
$\F_{3}$	2.3.a_g

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.81.a_agg	$2$	(not in LMFDB)
2.81.bk_ss	$2$	(not in LMFDB)
2.81.aj_a	$3$	(not in LMFDB)
2.81.s_jj	$3$	(not in LMFDB)