Abelian variety isogeny class 4.3.a_a_a_g over $\F_{3}$

• Label: 4.3.a_a_a_g
• Base field: $\F_{3}$
• Dimension: $4$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes

Invariants

Base field:	$\F_{3}$
Dimension:	$4$
L-polynomial:	$1 + 6 x^{4} + 81 x^{8}$
Frobenius angles:	$\pm0.152043361992$, $\pm0.347956638008$, $\pm0.652043361992$, $\pm0.847956638008$
Angle rank:	$1$ (numerical)
Number field:	8.0.3057647616.5
Galois group:	$D_4\times C_2$
Isomorphism classes:	260

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

Newton polygon

$p$-rank:	$0$
Slopes:	$[1/4, 1/4, 1/4, 1/4, 3/4, 3/4, 3/4, 3/4]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$88$	$7744$	$530200$	$59969536$	$3486901528$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$4$	$10$	$28$	$106$	$244$	$730$	$2188$	$7066$	$19684$	$59050$

Jacobians and polarizations

This isogeny class is principally polarizable, but it is unknown whether it contains a Jacobian.

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{3}$

The endomorphism algebra of this simple isogeny class is 8.0.3057647616.5.

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

The base change of $A$ to $\F_{3^{4}}$ is the simple isogeny class 4.81.y_uu_jxo_ehxu and its endomorphism algebra is the division algebra of dimension 16 over \(\Q(\sqrt{-2}) \) with the following ramification data at primes above $3$, and unramified at all archimedean places:

$v$	($ 3 $,\( \pi + 2 \))	($ 3 $,\( \pi + 1 \))
$\operatorname{inv}_v$	$1/4$	$3/4$

where $\pi$ is a root of $x^{2} + 2$.

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{3^{2}}$

  The base change of $A$ to $\F_{3^{2}}$ is the simple isogeny class 4.9.a_m_a_hq and its endomorphism algebra is the quaternion algebra over \(\Q(\sqrt{-2}, \sqrt{3})\) with the following ramification data at primes above $3$, and unramified at all archimedean places:

  $v$	($ 3 $,\( \pi + 2 \))	($ 3 $,\( \pi + 1 \))
  $\operatorname{inv}_v$	$1/2$	$1/2$

  where $\pi$ is a root of $x^{4} + 4x^{2} + 1$.

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
4.3.a_a_a_ag	$8$	(not in LMFDB)
4.3.a_ag_a_v	$24$	(not in LMFDB)
4.3.a_g_a_v	$24$	(not in LMFDB)