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

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

Invariants

Base field:	$\F_{2}$
Dimension:	$6$
L-polynomial:	$1 - x + x^{3} - 2 x^{4} - x^{5} + 11 x^{6} - 2 x^{7} - 8 x^{8} + 8 x^{9} - 32 x^{11} + 64 x^{12}$
Frobenius angles:	$\pm0.101847760735$, $\pm0.181850269129$, $\pm0.430688013037$, $\pm0.484816397538$, $\pm0.768514427402$, $\pm0.902645320296$
Angle rank:	$3$ (numerical)
Number field:	12.0.2011660281559296.2
Galois group:	$C_2 \times S_4$
Jacobians:	$0$
Cyclic group of points:	yes

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$6$
Slopes:	$[0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$39$	$3627$	$492804$	$12538539$	$785461989$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$2$	$4$	$11$	$12$	$22$	$115$	$184$	$260$	$479$	$1084$

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 $\F_{2^{3}}$.

Endomorphism algebra over $\F_{2}$

The endomorphism algebra of this simple isogeny class is 12.0.2011660281559296.2.

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

The base change of $A$ to $\F_{2^{3}}$ is 3.8.b_n_h^2 and its endomorphism algebra is $\mathrm{M}_{2}($6.0.14950512.2$)$

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
6.2.b_a_ab_ac_b_l	$2$	(not in LMFDB)
6.2.c_d_e_h_o_bd	$3$	(not in LMFDB)
6.2.ac_d_ae_h_ao_bd	$6$	(not in LMFDB)