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

• Label: 6.2.aj_bq_afd_mh_axm_bkq
• 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:	$( 1 - 2 x + 2 x^{2} )^{2}( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )( 1 - 2 x + 3 x^{2} - 4 x^{3} + 4 x^{4} )$
	$1 - 9 x + 42 x^{2} - 133 x^{3} + 319 x^{4} - 610 x^{5} + 952 x^{6} - 1220 x^{7} + 1276 x^{8} - 1064 x^{9} + 672 x^{10} - 288 x^{11} + 64 x^{12}$
Frobenius angles:	$\pm0.123548644961$, $\pm0.174442860055$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.456881978294$, $\pm0.546783656212$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2$	$13300$	$796328$	$23940000$	$2846407042$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-6$	$8$	$15$	$24$	$64$	$113$	$134$	$208$	$483$	$1048$

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.ac_d 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.ac_d : 4.0.1088.2.

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.adu_hrl. 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.adu_hrl : 4.0.1088.2.

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

  • 1.4.a^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
  • 2.4.b_ad : \(\Q(\sqrt{-3}, \sqrt{5})\).
  • 2.4.c_b : 4.0.1088.2.
• Endomorphism algebra over $\F_{2^{3}}$
  The base change of $A$ to $\F_{2^{3}}$ is 1.8.e^2 $\times$ 2.8.ac_p $\times$ 2.8.a_l. The endomorphism algebra for each factor is:

  • 1.8.e^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
  • 2.8.ac_p : 4.0.1088.2.
  • 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.ac_b. 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.ac_b : 4.0.1088.2.
• 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.ba_ld. 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.ba_ld : 4.0.1088.2.

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.af_o_abd_bz_ada_ei	$2$	(not in LMFDB)
6.2.af_o_az_bf_aba_y	$2$	(not in LMFDB)
6.2.ad_g_ah_h_c_ai	$2$	(not in LMFDB)
6.2.ab_c_af_h_ak_y	$2$	(not in LMFDB)
6.2.ab_c_ab_d_ag_q	$2$	(not in LMFDB)
6.2.b_c_b_d_g_q	$2$	(not in LMFDB)
6.2.b_c_f_h_k_y	$2$	(not in LMFDB)
6.2.d_g_h_h_ac_ai	$2$	(not in LMFDB)
6.2.f_o_z_bf_ba_y	$2$	(not in LMFDB)
6.2.f_o_bd_bz_da_ei	$2$	(not in LMFDB)
6.2.j_bq_fd_mh_xm_bkq	$2$	(not in LMFDB)
6.2.ag_s_abi_bx_acm_dk	$3$	(not in LMFDB)
6.2.ad_g_ah_b_o_aba	$3$	(not in LMFDB)
6.2.ad_g_ah_h_c_ai	$3$	(not in LMFDB)
6.2.a_a_c_b_c_k	$3$	(not in LMFDB)
6.2.d_g_l_n_o_w	$3$	(not in LMFDB)