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

• Label: 6.2.ac_g_ai_r_aba_bs
• Base field: $\F_{2}$
• Dimension: $6$
• $p$-rank: $4$
• 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:	$( 1 + 2 x + 2 x^{2} )( 1 - x + 2 x^{2} )^{3}( 1 - x - 2 x^{3} + 4 x^{4} )$
	$1 - 2 x + 6 x^{2} - 8 x^{3} + 17 x^{4} - 26 x^{5} + 44 x^{6} - 52 x^{7} + 68 x^{8} - 64 x^{9} + 96 x^{10} - 64 x^{11} + 64 x^{12}$
Frobenius angles:	$\pm0.139386741866$, $\pm0.384973271919$, $\pm0.384973271919$, $\pm0.384973271919$, $\pm0.686170398078$, $\pm0.750000000000$
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:	$4$
Slopes:	$[0, 0, 0, 0, 1/2, 1/2, 1/2, 1/2, 1, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$80$	$40960$	$356720$	$42598400$	$373212400$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$1$	$13$	$13$	$29$	$1$	$37$	$225$	$349$	$553$	$933$

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.ab^3 $\times$ 1.2.c $\times$ 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.ab^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$
• 1.2.c : \(\Q(\sqrt{-1}) \).
• 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.ab^3 $\times$ 1.16.i $\times$ 2.16.h_bo. The endomorphism algebra for each factor is:

• 1.16.ab^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$
• 1.16.i : 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 $\times$ 1.4.d^3 $\times$ 2.4.ab_e. The endomorphism algebra for each factor is:

  • 1.4.a : \(\Q(\sqrt{-1}) \).
  • 1.4.d^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$
  • 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.ag_w_ace_ej_ahi_ky	$2$	(not in LMFDB)
6.2.ae_m_aba_bz_adi_fc	$2$	(not in LMFDB)
6.2.ae_m_as_t_c_am	$2$	(not in LMFDB)
6.2.ac_g_am_z_abm_ci	$2$	(not in LMFDB)
6.2.ac_g_ae_j_c_m	$2$	(not in LMFDB)
6.2.a_e_ag_l_ao_bk	$2$	(not in LMFDB)
6.2.a_e_ac_l_ak_u	$2$	(not in LMFDB)
6.2.a_e_c_l_k_u	$2$	(not in LMFDB)
6.2.a_e_g_l_o_bk	$2$	(not in LMFDB)
6.2.c_g_e_j_ac_m	$2$	(not in LMFDB)
6.2.c_g_i_r_ba_bs	$2$	(not in LMFDB)
6.2.c_g_m_z_bm_ci	$2$	(not in LMFDB)
6.2.e_m_s_t_ac_am	$2$	(not in LMFDB)
6.2.e_m_ba_bz_di_fc	$2$	(not in LMFDB)
6.2.g_w_ce_ej_hi_ky	$2$	(not in LMFDB)
6.2.b_a_b_f_e_ae	$3$	(not in LMFDB)