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

• Label: 6.2.ac_d_ad_e_b_ag
• 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:	$( 1 - 2 x + 2 x^{2} )( 1 + x + 2 x^{2} )( 1 - x + 3 x^{2} - 2 x^{3} + 4 x^{4} )( 1 - 3 x^{2} + 4 x^{4} )$
	$1 - 2 x + 3 x^{2} - 3 x^{3} + 4 x^{4} + x^{5} - 6 x^{6} + 2 x^{7} + 16 x^{8} - 24 x^{9} + 48 x^{10} - 64 x^{11} + 64 x^{12}$
Frobenius angles:	$\pm0.115026728081$, $\pm0.250000000000$, $\pm0.306143893905$, $\pm0.570118980449$, $\pm0.615026728081$, $\pm0.884973271919$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$40$	$8800$	$307840$	$28160000$	$2683901000$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$1$	$7$	$10$	$23$	$61$	$64$	$57$	$319$	$550$	$1107$

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.b $\times$ 2.2.ab_d $\times$ 2.2.a_ad 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.b : \(\Q(\sqrt{-7}) \).
• 2.2.ab_d : 4.0.1025.1.
• 2.2.a_ad : \(\Q(i, \sqrt{7})\).

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

The base change of $A$ to $\F_{2^{4}}$ is 1.16.ab^3 $\times$ 1.16.i $\times$ 2.16.b_b. 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.b_b : 4.0.1025.1.

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{2^{2}}$
  The base change of $A$ to $\F_{2^{2}}$ is 1.4.ad^2 $\times$ 1.4.a $\times$ 1.4.d $\times$ 2.4.f_n. The endomorphism algebra for each factor is:

  • 1.4.ad^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$
  • 1.4.a : \(\Q(\sqrt{-1}) \).
  • 1.4.d : \(\Q(\sqrt{-7}) \).
  • 2.4.f_n : 4.0.1025.1.

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_j_al_g_h_as	$2$	(not in LMFDB)
6.2.ac_d_ab_a_ab_g	$2$	(not in LMFDB)
6.2.a_b_ab_c_b_c	$2$	(not in LMFDB)
6.2.a_b_b_c_ab_c	$2$	(not in LMFDB)
6.2.c_d_b_a_b_g	$2$	(not in LMFDB)
6.2.c_d_d_e_ab_ag	$2$	(not in LMFDB)
6.2.e_j_l_g_ah_as	$2$	(not in LMFDB)