Abelian variety isogeny class 3.2.ac_f_ag over $\F_{2}$

• Label: 3.2.ac_f_ag
• Base field: $\F_{2}$
• Dimension: $3$
• $p$-rank: $2$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{2}$
Dimension:	$3$
L-polynomial:	$( 1 - 2 x + 2 x^{2} )( 1 - x + 2 x^{2} )( 1 + x + 2 x^{2} )$
	$1 - 2 x + 5 x^{2} - 6 x^{3} + 10 x^{4} - 8 x^{5} + 8 x^{6}$
Frobenius angles:	$\pm0.250000000000$, $\pm0.384973271919$, $\pm0.615026728081$
Angle rank:	$1$ (numerical)
Jacobians:	$0$
Isomorphism classes:	12

This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.

Newton polygon

$p$-rank:	$2$
Slopes:	$[0, 0, 1/2, 1/2, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$8$	$320$	$728$	$6400$	$39688$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$1$	$11$	$13$	$23$	$41$	$47$	$113$	$287$	$481$	$911$

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 $\times$ 1.2.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 : \(\Q(\sqrt{-7}) \).
• 1.2.b : \(\Q(\sqrt{-7}) \).

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

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

• 1.16.ab^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$
• 1.16.i : 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 $\times$ 1.4.d^2. The endomorphism algebra for each factor is:

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

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
3.2.ae_l_as	$2$	3.4.g_v_bw
3.2.a_d_ac	$2$	3.4.g_v_bw
3.2.a_d_c	$2$	3.4.g_v_bw
3.2.c_f_g	$2$	3.4.g_v_bw
3.2.e_l_s	$2$	3.4.g_v_bw