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

• Label: 6.2.aj_bq_afb_lu_avy_bhw
• 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:	$( 1 - 2 x + 2 x^{2} )^{3}( 1 - 3 x + 6 x^{2} - 9 x^{3} + 12 x^{4} - 12 x^{5} + 8 x^{6} )$
	$1 - 9 x + 42 x^{2} - 131 x^{3} + 306 x^{4} - 570 x^{5} + 880 x^{6} - 1140 x^{7} + 1224 x^{8} - 1048 x^{9} + 672 x^{10} - 288 x^{11} + 64 x^{12}$
Frobenius angles:	$\pm0.147012170705$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.341962716420$, $\pm0.600633654388$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3$	$19125$	$1127061$	$88453125$	$3233980083$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-6$	$8$	$21$	$44$	$69$	$59$	$78$	$236$	$498$	$983$

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 $\times$ 3.2.ad_g_aj 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 : $\mathrm{M}_{3}($\(\Q(\sqrt{-1}) \)$)$
• 3.2.ad_g_aj : 6.0.465831.1.

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

The base change of $A$ to $\F_{2^{4}}$ is 1.16.i^3 $\times$ 3.16.d_bq_db. The endomorphism algebra for each factor is:

• 1.16.i^3 : $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 3.16.d_bq_db : 6.0.465831.1.

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^3 $\times$ 3.4.d_g_h. The endomorphism algebra for each factor is:

  • 1.4.a^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-1}) \)$)$
  • 3.4.d_g_h : 6.0.465831.1.

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.af_o_abb_bu_acw_ei	$2$	(not in LMFDB)
6.2.ad_g_af_g_ag_q	$2$	(not in LMFDB)
6.2.ab_c_ad_k_ak_q	$2$	(not in LMFDB)
6.2.b_c_d_k_k_q	$2$	(not in LMFDB)
6.2.d_g_f_g_g_q	$2$	(not in LMFDB)
6.2.f_o_bb_bu_cw_ei	$2$	(not in LMFDB)
6.2.j_bq_fb_lu_vy_bhw	$2$	(not in LMFDB)
6.2.ad_g_af_a_m_au	$3$	(not in LMFDB)