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

• Label: 6.2.af_j_ah_d_ab_ab
• 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:	$1 - 5 x + 9 x^{2} - 7 x^{3} + 3 x^{4} - x^{5} - x^{6} - 2 x^{7} + 12 x^{8} - 56 x^{9} + 144 x^{10} - 160 x^{11} + 64 x^{12}$
Frobenius angles:	$\pm0.0211347627837$, $\pm0.0689543221748$, $\pm0.137284443974$, $\pm0.422998729689$, $\pm0.640382893603$, $\pm0.878277619927$
Angle rank:	$3$ (numerical)
Number field:	12.0.16135268698129.1
Galois group:	$C_2^2 \times A_4$
Jacobians:	$0$
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1$	$463$	$58801$	$7701079$	$668807021$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-2$	$-2$	$-2$	$2$	$18$	$58$	$131$	$338$	$412$	$1018$

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 $\F_{2^{7}}$.

Endomorphism algebra over $\F_{2}$

The endomorphism algebra of this simple isogeny class is 12.0.16135268698129.1.

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

The base change of $A$ to $\F_{2^{7}}$ is 3.128.b_acv_bfj^2 and its endomorphism algebra is $\mathrm{M}_{2}($6.0.573839.1$)$

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.f_j_h_d_b_ab	$2$	(not in LMFDB)
6.2.c_c_a_d_g_n	$7$	(not in LMFDB)
6.2.c_c_h_k_n_bb	$7$	(not in LMFDB)