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

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

Invariants

Base field:	$\F_{2}$
Dimension:	$6$
L-polynomial:	$1 - x - 2 x^{2} + 3 x^{3} - x^{5} + 3 x^{6} - 2 x^{7} + 24 x^{9} - 32 x^{10} - 32 x^{11} + 64 x^{12}$
Frobenius angles:	$\pm0.111704523873$, $\pm0.131583750300$, $\pm0.342108384675$, $\pm0.554962142793$, $\pm0.798250416967$, $\pm0.991224948659$
Angle rank:	$3$ (numerical)
Number field:	12.0.7974955795601664.1
Galois group:	12T48
Jacobians:	$0$

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

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$25$	$1075$	$422500$	$12926875$	$1398731875$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$2$	$0$	$11$	$12$	$42$	$75$	$156$	$292$	$767$	$1000$

Jacobians and polarizations

This isogeny class is not principally polarizable, and therefore 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 12.0.7974955795601664.1.

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

The base change of $A$ to $\F_{2^{3}}$ is 3.8.b_d_bt^2 and its endomorphism algebra is $\mathrm{M}_{2}($6.0.3307504.1$)$

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_ac_ad_a_b_d	$2$	(not in LMFDB)
6.2.c_h_m_bb_bm_cr	$3$	(not in LMFDB)
6.2.ac_h_am_bb_abm_cr	$6$	(not in LMFDB)