Abelian variety isogeny class 2.3.a_af over $\F_{3}$

• Label: 2.3.a_af
• Base field: $\F_{3}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{3}$
Dimension:	$2$
L-polynomial:	$1 - 5 x^{2} + 9 x^{4}$
Frobenius angles:	$\pm0.0932147493387$, $\pm0.906785250661$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(i, \sqrt{11})\)
Galois group:	$C_2^2$
Jacobians:	$0$
Isomorphism classes:	1

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5$	$25$	$740$	$5625$	$59525$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$4$	$0$	$28$	$68$	$244$	$750$	$2188$	$6788$	$19684$	$60000$

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 $\F_{3^{2}}$.

Endomorphism algebra over $\F_{3}$

The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{11})\).

Endomorphism algebra over $\overline{\F}_{3}$

The base change of $A$ to $\F_{3^{2}}$ is 1.9.af^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$

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
2.3.ac_h	$4$	2.81.ao_id
2.3.a_f	$4$	2.81.ao_id
2.3.c_h	$4$	2.81.ao_id