Abelian variety isogeny class 4.2.a_ae_a_k over $\F_{2}$

• Label: 4.2.a_ae_a_k
• Base field: $\F_{2}$
• Dimension: $4$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{2}$
Dimension:	$4$
L-polynomial:	$1 - 4 x^{2} + 10 x^{4} - 16 x^{6} + 16 x^{8}$
Frobenius angles:	$\pm0.0872214300276$, $\pm0.226608171894$, $\pm0.773391828106$, $\pm0.912778569972$
Angle rank:	$2$ (numerical)
Number field:	8.0.1212153856.10
Galois group:	$C_2^2:C_4$
Jacobians:	$0$
Isomorphism classes:	2

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

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$7$	$49$	$4753$	$108241$	$1064497$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$3$	$-3$	$9$	$25$	$33$	$81$	$129$	$241$	$513$	$1057$

Jacobians and polarizations

This isogeny class does not contain a Jacobian, and it is unknown whether it is principally polarizable.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{2}}$.

Endomorphism algebra over $\F_{2}$

The endomorphism algebra of this simple isogeny class is 8.0.1212153856.10.

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

The base change of $A$ to $\F_{2^{2}}$ is the simple isogeny class 4.4.ai_bk_aei_ka and its endomorphism algebra is the quaternion algebra over 4.0.1088.2 with the following ramification data at primes above $2$, and unramified at all archimedean places:

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

where $\pi$ is a root of $x^{4} - 2x^{3} + 5x^{2} - 4x + 2$.

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
4.2.a_e_a_k	$4$	(not in LMFDB)
4.2.ae_g_ae_c	$16$	(not in LMFDB)
4.2.a_c_ae_c	$16$	(not in LMFDB)