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

• Label: 6.2.aj_bt_afx_pa_abdu_bvc
• 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:	$( 1 - 2 x + 2 x^{2} )^{3}( 1 - x + 2 x^{2} )^{3}$
	$1 - 9 x + 45 x^{2} - 153 x^{3} + 390 x^{4} - 774 x^{5} + 1224 x^{6} - 1548 x^{7} + 1560 x^{8} - 1224 x^{9} + 720 x^{10} - 288 x^{11} + 64 x^{12}$
Frobenius angles:	$\pm0.250000000000$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.384973271919$, $\pm0.384973271919$, $\pm0.384973271919$
Angle rank:	$1$ (numerical)
Jacobians:	$0$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$8$	$64000$	$6028568$	$64000000$	$733870808$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-6$	$14$	$36$	$38$	$24$	$38$	$120$	$254$	$432$	$854$

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^3 $\times$ 1.2.ab^3 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^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$

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

The base change of $A$ to $\F_{2^{4}}$ is 1.16.ab^3 $\times$ 1.16.i^3. The endomorphism algebra for each factor is:

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

  • 1.4.a^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-1}) \)$)$
  • 1.4.d^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$

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_bd_adf_he_ana_ua	$2$	(not in LMFDB)
6.2.af_r_abp_di_afu_iy	$2$	(not in LMFDB)
6.2.af_r_abl_co_adq_fg	$2$	(not in LMFDB)
6.2.ad_j_at_bq_aco_ea	$2$	(not in LMFDB)
6.2.ad_j_ap_be_abq_cu	$2$	(not in LMFDB)
6.2.ab_f_af_s_ao_bo	$2$	(not in LMFDB)
6.2.ab_f_ab_o_ag_bo	$2$	(not in LMFDB)
6.2.b_f_b_o_g_bo	$2$	(not in LMFDB)
6.2.b_f_f_s_o_bo	$2$	(not in LMFDB)
6.2.d_j_p_be_bq_cu	$2$	(not in LMFDB)
6.2.d_j_t_bq_co_ea	$2$	(not in LMFDB)
6.2.f_r_bl_co_dq_fg	$2$	(not in LMFDB)
6.2.f_r_bp_di_fu_iy	$2$	(not in LMFDB)
6.2.h_bd_df_he_na_ua	$2$	(not in LMFDB)
6.2.j_bt_fx_pa_bdu_bvc	$2$	(not in LMFDB)
6.2.ag_s_abb_g_co_afo	$3$	(not in LMFDB)
6.2.ad_j_aj_g_y_abk	$3$	(not in LMFDB)
6.2.a_a_j_a_a_bk	$3$	(not in LMFDB)