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

• Label: 6.2.ai_be_acn_de_aby_q
• Base field: $\F_{2}$
• Dimension: $6$
• $p$-rank: $3$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{2}$
Dimension:	$6$
L-polynomial:	$( 1 - 2 x + 2 x^{2} )^{3}( 1 - 2 x + 3 x^{3} - 8 x^{5} + 8 x^{6} )$
	$1 - 8 x + 30 x^{2} - 65 x^{3} + 82 x^{4} - 50 x^{5} + 16 x^{6} - 100 x^{7} + 328 x^{8} - 520 x^{9} + 480 x^{10} - 256 x^{11} + 64 x^{12}$
Frobenius angles:	$\pm0.0889496890695$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.297004294965$, $\pm0.823081333977$
Angle rank:	$3$ (numerical)
Jacobians:	$0$

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2$	$4000$	$1388504$	$109000000$	$1608753982$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-5$	$1$	$22$	$49$	$45$	$70$	$51$	$177$	$490$	$1101$

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 $\F_{2^{4}}$.

Endomorphism algebra over $\F_{2}$

The isogeny class factors as 1.2.ac^3 $\times$ 3.2.ac_a_d and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.2.ac^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-1}) \)$)$
• 3.2.ac_a_d : 6.0.1539727.2.

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

The base change of $A$ to $\F_{2^{4}}$ is 1.16.i^3 $\times$ 3.16.i_bo_fn. The endomorphism algebra for each factor is:

• 1.16.i^3 : $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 3.16.i_bo_fn : 6.0.1539727.2.

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{2^{2}}$
  The base change of $A$ to $\F_{2^{2}}$ is 1.4.a^3 $\times$ 3.4.ae_m_az. The endomorphism algebra for each factor is:

  • 1.4.a^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-1}) \)$)$
  • 3.4.ae_m_az : 6.0.1539727.2.

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.ae_g_ab_ac_as_bw	$2$	(not in LMFDB)
6.2.ae_g_b_ak_c_q	$2$	(not in LMFDB)
6.2.a_ac_ab_k_ac_aq	$2$	(not in LMFDB)
6.2.a_ac_b_k_c_aq	$2$	(not in LMFDB)
6.2.e_g_ab_ak_ac_q	$2$	(not in LMFDB)
6.2.e_g_b_ac_s_bw	$2$	(not in LMFDB)
6.2.i_be_cn_de_by_q	$2$	(not in LMFDB)
6.2.ac_a_h_ai_ai_bc	$3$	(not in LMFDB)