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

• Label: 6.2.ai_bg_adh_gq_ale_qq
• 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:	$( 1 - 2 x + 2 x^{2} )^{2}( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )( 1 - x - 2 x^{3} + 4 x^{4} )$
	$1 - 8 x + 32 x^{2} - 85 x^{3} + 172 x^{4} - 290 x^{5} + 432 x^{6} - 580 x^{7} + 688 x^{8} - 680 x^{9} + 512 x^{10} - 256 x^{11} + 64 x^{12}$
Frobenius angles:	$\pm0.123548644961$, $\pm0.139386741866$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.456881978294$, $\pm0.686170398078$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2$	$7600$	$333944$	$44460000$	$2264848282$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-5$	$5$	$10$	$33$	$55$	$86$	$177$	$241$	$442$	$1165$

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^2 $\times$ 2.2.ad_f $\times$ 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^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
• 2.2.ad_f : \(\Q(\sqrt{-3}, \sqrt{5})\).
• 2.2.ab_a : 4.0.2312.1.

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

The base change of $A$ to $\F_{2^{12}}$ is 1.4096.h^2 $\times$ 1.4096.ey^2 $\times$ 2.4096.agf_vki. The endomorphism algebra for each factor is:

• 1.4096.h^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$
• 1.4096.ey^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 2.4096.agf_vki : 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^2 $\times$ 2.4.ab_e $\times$ 2.4.b_ad. The endomorphism algebra for each factor is:

  • 1.4.a^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
  • 2.4.ab_e : 4.0.2312.1.
  • 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^2 $\times$ 2.8.ah_y $\times$ 2.8.a_l. The endomorphism algebra for each factor is:

  • 1.8.e^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
  • 2.8.ah_y : 4.0.2312.1.
  • 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.i^2 $\times$ 2.16.ah_bh $\times$ 2.16.h_bo. The endomorphism algebra for each factor is:

  • 1.16.i^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 2.16.ah_bh : \(\Q(\sqrt{-3}, \sqrt{5})\).
  • 2.16.h_bo : 4.0.2312.1.
• Endomorphism algebra over $\F_{2^{6}}$
  The base change of $A$ to $\F_{2^{6}}$ is 1.64.a^2 $\times$ 1.64.l^2 $\times$ 2.64.ab_adc. The endomorphism algebra for each factor is:

  • 1.64.a^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
  • 1.64.l^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$
  • 2.64.ab_adc : 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_s_abf_bc_c_abg	$2$	(not in LMFDB)
6.2.ae_i_an_y_abq_cm	$2$	(not in LMFDB)
6.2.ac_c_ab_e_ac_a	$2$	(not in LMFDB)
6.2.ac_c_b_a_ag_q	$2$	(not in LMFDB)
6.2.a_a_af_e_ac_q	$2$	(not in LMFDB)
6.2.a_a_f_e_c_q	$2$	(not in LMFDB)
6.2.c_c_ab_a_g_q	$2$	(not in LMFDB)
6.2.c_c_b_e_c_a	$2$	(not in LMFDB)
6.2.e_i_n_y_bq_cm	$2$	(not in LMFDB)
6.2.g_s_bf_bc_ac_abg	$2$	(not in LMFDB)
6.2.i_bg_dh_gq_le_qq	$2$	(not in LMFDB)
6.2.af_l_an_q_abm_cu	$3$	(not in LMFDB)
6.2.ac_c_ab_ac_e_a	$3$	(not in LMFDB)
6.2.ac_c_ab_e_ac_a	$3$	(not in LMFDB)
6.2.b_ab_ab_e_e_a	$3$	(not in LMFDB)
6.2.e_i_l_k_e_a	$3$	(not in LMFDB)