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

• Label: 6.2.aj_bo_aek_jc_apc_we
• Base field: $\F_{2}$
• Dimension: $6$
• $p$-rank: $1$
• 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:	$( 1 - 2 x + 2 x^{2} )^{4}( 1 - x - 2 x^{3} + 4 x^{4} )$
	$1 - 9 x + 40 x^{2} - 114 x^{3} + 236 x^{4} - 392 x^{5} + 576 x^{6} - 784 x^{7} + 944 x^{8} - 912 x^{9} + 640 x^{10} - 288 x^{11} + 64 x^{12}$
Frobenius angles:	$\pm0.139386741866$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.686170398078$
Angle rank:	$2$ (numerical)
Jacobians:	$0$

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2$	$10000$	$742586$	$162500000$	$3961716922$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-6$	$4$	$18$	$56$	$74$	$64$	$106$	$160$	$378$	$1104$

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^4 $\times$ 2.2.ab_a 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^4 : $\mathrm{M}_{4}($\(\Q(\sqrt{-1}) \)$)$
• 2.2.ab_a : 4.0.2312.1.

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

The base change of $A$ to $\F_{2^{4}}$ is 1.16.i^4 $\times$ 2.16.h_bo. The endomorphism algebra for each factor is:

• 1.16.i^4 : $\mathrm{M}_{4}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 2.16.h_bo : 4.0.2312.1.

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^4 $\times$ 2.4.ab_e. The endomorphism algebra for each factor is:

  • 1.4.a^4 : $\mathrm{M}_{4}($\(\Q(\sqrt{-1}) \)$)$
  • 2.4.ab_e : 4.0.2312.1.

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.ah_y_abu_bs_i_acm	$2$	(not in LMFDB)
6.2.af_m_as_bc_ace_ds	$2$	(not in LMFDB)
6.2.ad_e_c_ae_ai_bg	$2$	(not in LMFDB)
6.2.ab_a_ac_m_ai_a	$2$	(not in LMFDB)
6.2.b_a_c_m_i_a	$2$	(not in LMFDB)
6.2.d_e_ac_ae_i_bg	$2$	(not in LMFDB)
6.2.f_m_s_bc_ce_ds	$2$	(not in LMFDB)
6.2.h_y_bu_bs_ai_acm	$2$	(not in LMFDB)
6.2.j_bo_ek_jc_pc_we	$2$	(not in LMFDB)
6.2.ad_e_a_ae_e_a	$3$	(not in LMFDB)
6.2.d_e_g_i_e_a	$3$	(not in LMFDB)