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

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

Invariants

Base field:	$\F_{2}$
Dimension:	$6$
L-polynomial:	$1 - 2 x^{3} - 16 x^{9} + 64 x^{12}$
Frobenius angles:	$\pm0.0464622472888$, $\pm0.228723466026$, $\pm0.437943200641$, $\pm0.620204419378$, $\pm0.713128913955$, $\pm0.895390132693$
Angle rank:	$2$ (numerical)
Number field:	12.0.105084467666362368.1
Galois group:	12T78
Jacobians:	$1$

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$47$	$3901$	$103823$	$17246321$	$1083261217$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$3$	$5$	$3$	$17$	$33$	$53$	$129$	$257$	$345$	$1025$

Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is not hyperelliptic), and hence is principally polarizable:

• $x^{10} + x^{6} y^{4} + x^{9} + x^{7} y^{2} + x^{6} y^{2} + x^{3} y^{4} + x^{2} y^{5} + x^{2} y^{4} + x^{5} + x^{4} y + x^{3} y^{2} + x y^{4} + x^{4} + y^{4} + x^{2} y + x y^{2} + y^{3} + y^{2} + x + y + 1=0$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{3}}$.

Endomorphism algebra over $\F_{2}$

The endomorphism algebra of this simple isogeny class is 12.0.105084467666362368.1.

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

The base change of $A$ to $\F_{2^{3}}$ is the simple isogeny class 6.8.ag_m_ace_ou_abky_chc and its endomorphism algebra is the division algebra of dimension 9 over 4.0.2312.1 with the following ramification data at primes above $2$, and unramified at all archimedean places:

$v$	($ 2 $,\( \pi + 1 \))	($ 2 $,\( \frac{1}{2} \pi^{3} + \frac{1}{2} \pi^{2} + 1 \))	($ 2 $,\( \frac{1}{2} \pi^{3} + \frac{3}{2} \pi^{2} \))
$\operatorname{inv}_v$	$1/3$	$0$	$2/3$

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

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_a_c_a_a_a	$2$	(not in LMFDB)
6.2.a_a_c_a_a_a	$6$	(not in LMFDB)