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

• Label: 2.3.a_ac
• 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: yes

Invariants

Base field:	$\F_{3}$
Dimension:	$2$
L-polynomial:	$1 - 2 x^{2} + 9 x^{4}$
Frobenius angles:	$\pm0.195913276015$, $\pm0.804086723985$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\zeta_{8})\)
Galois group:	$C_2^2$
Jacobians:	$2$
Isomorphism classes:	5

This isogeny class is simple but not geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$8$	$64$	$776$	$9216$	$58568$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$4$	$6$	$28$	$110$	$244$	$822$	$2188$	$6494$	$19684$	$58086$

Jacobians and polarizations

This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=x^5+2x$
• $y^2=x^5+x^4+2x^2+x+2$

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 \(\Q(\zeta_{8})\).

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

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

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.ae_k	$4$	2.81.bc_nu
2.3.a_c	$4$	2.81.bc_nu
2.3.e_k	$4$	2.81.bc_nu