Abelian variety isogeny class 6.2.a_ac_a_c_a_a over $\F_{2}$

• Label: 6.2.a_ac_a_c_a_a
• Base field: $\F_{2}$
• Dimension: $6$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{2}$
Dimension:	$6$
L-polynomial:	$1 - 2 x^{2} + 2 x^{4} + 8 x^{8} - 32 x^{10} + 64 x^{12}$
Frobenius angles:	$\pm0.0764351249463$, $\pm0.189398850982$, $\pm0.387036273964$, $\pm0.612963726036$, $\pm0.810601149018$, $\pm0.923564875054$
Angle rank:	$2$ (numerical)
Number field:	12.0.319794774016000000.1
Galois group:	12T48
Jacobians:	$0$

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

Newton polygon

$p$-rank:	$0$
Slopes:	$[1/4, 1/4, 1/4, 1/4, 1/2, 1/2, 1/2, 1/2, 3/4, 3/4, 3/4, 3/4]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$41$	$1681$	$282941$	$19971961$	$999281561$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$3$	$1$	$9$	$17$	$33$	$73$	$129$	$337$	$513$	$881$

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 12.0.319794774016000000.1.

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

The base change of $A$ to $\F_{2^{2}}$ is the simple isogeny class 6.4.ae_i_ai_u_ads_lc and its endomorphism algebra is the quaternion algebra over 6.0.17672000.2 with the following ramification data at primes above $2$, and unramified at all archimedean places:

$v$	($ 2 $,\( \frac{1}{16} \pi^{5} + \frac{5}{16} \pi^{3} + \frac{7}{4} \pi^{2} + \frac{5}{4} \pi + 1 \))	($ 2 $,\( \frac{61}{32} \pi^{5} + \frac{3}{16} \pi^{4} + \frac{45}{32} \pi^{3} + \frac{1}{16} \pi^{2} + \frac{5}{4} \pi + 1 \))	($ 2 $,\( \pi + 1 \))
$\operatorname{inv}_v$	$1/2$	$0$	$1/2$

where $\pi$ is a root of $x^{6} + 5x^{4} - 4x^{3} + 20x^{2} + 64$.

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
6.2.a_c_a_c_a_a	$4$	(not in LMFDB)