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

• Label: 6.2.aj_bp_aes_ki_ase_bbk
• Base field: $\F_{2}$
• Dimension: $6$
• $p$-rank: $2$
• 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:	$( 1 - 2 x + 2 x^{2} )^{4}( 1 - x + x^{2} - 2 x^{3} + 4 x^{4} )$
	$1 - 9 x + 41 x^{2} - 122 x^{3} + 268 x^{4} - 472 x^{5} + 712 x^{6} - 944 x^{7} + 1072 x^{8} - 976 x^{9} + 656 x^{10} - 288 x^{11} + 64 x^{12}$
Frobenius angles:	$\pm0.197201053961$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.652365995579$
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:	$2$
Slopes:	$[0, 0, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3$	$16875$	$1028196$	$179296875$	$5094847083$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-6$	$6$	$21$	$58$	$84$	$63$	$78$	$146$	$309$	$966$

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_b 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_b : 4.0.2873.1.

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

The base change of $A$ to $\F_{2^{4}}$ is 1.16.i^4 $\times$ 2.16.j_bx. 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.j_bx : 4.0.2873.1.

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.b_f. The endomorphism algebra for each factor is:

  • 1.4.a^4 : $\mathrm{M}_{4}($\(\Q(\sqrt{-1}) \)$)$
  • 2.4.b_f : 4.0.2873.1.

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_z_acc_cy_acu_cu	$2$	(not in LMFDB)
6.2.af_n_aw_bk_acm_ea	$2$	(not in LMFDB)
6.2.ad_f_ac_e_aq_bo	$2$	(not in LMFDB)
6.2.ab_b_ac_m_ai_i	$2$	(not in LMFDB)
6.2.b_b_c_m_i_i	$2$	(not in LMFDB)
6.2.d_f_c_e_q_bo	$2$	(not in LMFDB)
6.2.f_n_w_bk_cm_ea	$2$	(not in LMFDB)
6.2.h_z_cc_cy_cu_cu	$2$	(not in LMFDB)
6.2.j_bp_es_ki_se_bbk	$2$	(not in LMFDB)
6.2.ad_f_ac_ac_i_ai	$3$	(not in LMFDB)
6.2.d_f_k_q_u_bc	$3$	(not in LMFDB)