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

• Label: 6.2.ak_ca_agz_ry_abju_ceq
• 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:	$( 1 - x + 2 x^{2} )( 1 - 2 x + 2 x^{2} )^{3}( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )$
	$1 - 10 x + 52 x^{2} - 181 x^{3} + 466 x^{4} - 930 x^{5} + 1472 x^{6} - 1860 x^{7} + 1864 x^{8} - 1448 x^{9} + 832 x^{10} - 320 x^{11} + 64 x^{12}$
Frobenius angles:	$\pm0.123548644961$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.384973271919$, $\pm0.456881978294$
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:	$3$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2$	$19000$	$2337608$	$42750000$	$1457127782$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-7$	$9$	$26$	$33$	$43$	$78$	$133$	$209$	$422$	$1029$

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 $\F_{2^{12}}$.

Endomorphism algebra over $\F_{2}$

The isogeny class factors as 1.2.ac^3 $\times$ 1.2.ab $\times$ 2.2.ad_f 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 : $\mathrm{M}_{3}($\(\Q(\sqrt{-1}) \)$)$
• 1.2.ab : \(\Q(\sqrt{-7}) \).
• 2.2.ad_f : \(\Q(\sqrt{-3}, \sqrt{5})\).

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

The base change of $A$ to $\F_{2^{12}}$ is 1.4096.h^2 $\times$ 1.4096.bv $\times$ 1.4096.ey^3. The endomorphism algebra for each factor is:

• 1.4096.h^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$
• 1.4096.bv : \(\Q(\sqrt{-7}) \).
• 1.4096.ey^3 : $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.

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 $\times$ 1.4.d $\times$ 2.4.b_ad. The endomorphism algebra for each factor is:

  • 1.4.a^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-1}) \)$)$
  • 1.4.d : \(\Q(\sqrt{-7}) \).
  • 2.4.b_ad : \(\Q(\sqrt{-3}, \sqrt{5})\).
• Endomorphism algebra over $\F_{2^{3}}$
  The base change of $A$ to $\F_{2^{3}}$ is 1.8.e^3 $\times$ 1.8.f $\times$ 2.8.a_l. The endomorphism algebra for each factor is:

  • 1.8.e^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-1}) \)$)$
  • 1.8.f : \(\Q(\sqrt{-7}) \).
  • 2.8.a_l : \(\Q(\sqrt{-3}, \sqrt{5})\).
• Endomorphism algebra over $\F_{2^{4}}$
  The base change of $A$ to $\F_{2^{4}}$ is 1.16.ab $\times$ 1.16.i^3 $\times$ 2.16.ah_bh. The endomorphism algebra for each factor is:

  • 1.16.ab : \(\Q(\sqrt{-7}) \).
  • 1.16.i^3 : $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 2.16.ah_bh : \(\Q(\sqrt{-3}, \sqrt{5})\).
• Endomorphism algebra over $\F_{2^{6}}$
  The base change of $A$ to $\F_{2^{6}}$ is 1.64.aj $\times$ 1.64.a^3 $\times$ 1.64.l^2. The endomorphism algebra for each factor is:

  • 1.64.aj : \(\Q(\sqrt{-7}) \).
  • 1.64.a^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-1}) \)$)$
  • 1.64.l^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$

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.ai_bi_adv_io_apq_ya	$2$	(not in LMFDB)
6.2.ag_u_abt_da_aek_ge	$2$	(not in LMFDB)
6.2.ae_k_at_bi_acc_dc	$2$	(not in LMFDB)
6.2.ae_k_an_k_g_aq	$2$	(not in LMFDB)
6.2.ac_e_af_k_as_bg	$2$	(not in LMFDB)
6.2.ac_e_ad_g_c_a	$2$	(not in LMFDB)
6.2.a_c_ad_g_ag_q	$2$	(not in LMFDB)
6.2.a_c_d_g_g_q	$2$	(not in LMFDB)
6.2.c_e_d_g_ac_a	$2$	(not in LMFDB)
6.2.c_e_f_k_s_bg	$2$	(not in LMFDB)
6.2.e_k_n_k_ag_aq	$2$	(not in LMFDB)
6.2.e_k_t_bi_cc_dc	$2$	(not in LMFDB)
6.2.g_u_bt_da_ek_ge	$2$	(not in LMFDB)
6.2.i_bi_dv_io_pq_ya	$2$	(not in LMFDB)
6.2.k_ca_gz_ry_bju_ceq	$2$	(not in LMFDB)
6.2.ah_z_acd_de_adm_ea	$3$	(not in LMFDB)
6.2.ae_k_an_e_y_aca	$3$	(not in LMFDB)
6.2.ab_b_f_ac_a_u	$3$	(not in LMFDB)
6.2.c_e_l_q_y_bs	$3$	(not in LMFDB)