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

• Label: 6.2.ag_x_acj_fa_ait_ni
• Base field: $\F_{2}$
• Dimension: $6$
• $p$-rank: $5$
• 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:	$( 1 - 2 x + 2 x^{2} )( 1 - x + 2 x^{2} )^{3}( 1 - x + x^{2} - 2 x^{3} + 4 x^{4} )$
	$1 - 6 x + 23 x^{2} - 61 x^{3} + 130 x^{4} - 227 x^{5} + 346 x^{6} - 454 x^{7} + 520 x^{8} - 488 x^{9} + 368 x^{10} - 192 x^{11} + 64 x^{12}$
Frobenius angles:	$\pm0.197201053961$, $\pm0.250000000000$, $\pm0.384973271919$, $\pm0.384973271919$, $\pm0.384973271919$, $\pm0.652365995579$
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:	$5$
Slopes:	$[0, 0, 0, 0, 0, 1/2, 1/2, 1, 1, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$24$	$69120$	$1284192$	$47001600$	$787132104$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-3$	$15$	$24$	$31$	$27$	$36$	$165$	$335$	$420$	$795$

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 $\times$ 1.2.ab^3 $\times$ 2.2.ab_b 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 : \(\Q(\sqrt{-1}) \).
• 1.2.ab^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$
• 2.2.ab_b : 4.0.2873.1.

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

The base change of $A$ to $\F_{2^{4}}$ is 1.16.ab^3 $\times$ 1.16.i $\times$ 2.16.j_bx. The endomorphism algebra for each factor is:

• 1.16.ab^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$
• 1.16.i : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 2.16.j_bx : 4.0.2873.1.

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 $\times$ 1.4.d^3 $\times$ 2.4.b_f. The endomorphism algebra for each factor is:

  • 1.4.a : \(\Q(\sqrt{-1}) \).
  • 1.4.d^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$
  • 2.4.b_f : 4.0.2873.1.

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.ae_n_abd_ci_adx_gc	$2$	(not in LMFDB)
6.2.ae_n_ax_bk_abj_by	$2$	(not in LMFDB)
6.2.ac_h_an_be_abr_cw	$2$	(not in LMFDB)
6.2.ac_h_aj_w_abb_cc	$2$	(not in LMFDB)
6.2.ac_h_ah_s_an_bm	$2$	(not in LMFDB)
6.2.a_f_af_q_an_bu	$2$	(not in LMFDB)
6.2.a_f_ab_q_af_bi	$2$	(not in LMFDB)
6.2.a_f_b_q_f_bi	$2$	(not in LMFDB)
6.2.a_f_f_q_n_bu	$2$	(not in LMFDB)
6.2.c_h_h_s_n_bm	$2$	(not in LMFDB)
6.2.c_h_j_w_bb_cc	$2$	(not in LMFDB)
6.2.c_h_n_be_br_cw	$2$	(not in LMFDB)
6.2.e_n_x_bk_bj_by	$2$	(not in LMFDB)
6.2.e_n_bd_ci_dx_gc	$2$	(not in LMFDB)
6.2.g_x_cj_fa_it_ni	$2$	(not in LMFDB)
6.2.ad_f_ab_af_n_ao	$3$	(not in LMFDB)