Abelian variety isogeny class 4.4.ag_s_abk_cq over $\F_{2^{2}}$

• Label: 4.4.ag_s_abk_cq
• Base field: $\F_{2^{2}}$
• Dimension: $4$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{2^{2}}$
Dimension:	$4$
L-polynomial:	$1 - 6 x + 18 x^{2} - 36 x^{3} + 68 x^{4} - 144 x^{5} + 288 x^{6} - 384 x^{7} + 256 x^{8}$
Frobenius angles:	$\pm0.126451355039$, $\pm0.206881978294$, $\pm0.373548644961$, $\pm0.706881978294$
Angle rank:	$1$ (numerical)
Number field:	8.0.12960000.1
Galois group:	$C_2^3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$61$	$73261$	$16776769$	$5367174121$	$1125584940601$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-1$	$17$	$65$	$313$	$1049$	$4097$	$17009$	$66081$	$262145$	$1048577$

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 $\F_{2^{24}}$.

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

The endomorphism algebra of this simple isogeny class is 8.0.12960000.1.

Endomorphism algebra over $\overline{\F}_{2^{2}}$

The base change of $A$ to $\F_{2^{24}}$ is the simple isogeny class 4.16777216.acqy_ftmswm_alhdndwaa_mgnaawzgjjs and its endomorphism algebra is the division algebra of dimension 16 over \(\Q(\sqrt{-15}) \) with the following ramification data at primes above $2$, and unramified at all archimedean places:

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

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

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{2^{4}}$

  The base change of $A$ to $\F_{2^{4}}$ is the simple isogeny class 4.16.a_bc_a_ui and its endomorphism algebra is 8.0.12960000.1.

• Endomorphism algebra over $\F_{2^{6}}$

  The base change of $A$ to $\F_{2^{6}}$ is the simple isogeny class 4.64.a_a_a_arg and its endomorphism algebra is 8.0.12960000.1.

• Endomorphism algebra over $\F_{2^{8}}$

  The base change of $A$ to $\F_{2^{8}}$ is the simple isogeny class 4.256.ce_csu_cmyq_buecm and its endomorphism algebra is the quaternion algebra over \(\Q(\sqrt{-3}, \sqrt{5})\) with the following ramification data at primes above $2$, and unramified at all archimedean places:

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

  where $\pi$ is a root of $x^{4} - x^{3} + 2x^{2} + x + 1$.
• Endomorphism algebra over $\F_{2^{12}}$

  The base change of $A$ to $\F_{2^{12}}$ is the simple isogeny class 4.4096.a_abim_a_cvwnqm and its endomorphism algebra is the quaternion algebra over \(\Q(i, \sqrt{15})\) with the following ramification data at primes above $2$, and unramified at all archimedean places:

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

  where $\pi$ is a root of $x^{4} - 7x^{2} + 16$.

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
4.4.g_s_bk_cq	$2$	(not in LMFDB)
4.4.a_a_a_abc	$3$	(not in LMFDB)
4.4.g_s_bk_cq	$3$	(not in LMFDB)