Abelian variety isogeny class 4.2.ae_h_ag_e over F2\F_{2}

• Label: 4.2.ae_h_ag_e
• Base field: $\F_{2}$
• Dimension: $4$
• pp-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:	$4$
L-polynomial:	$( 1 - 2 x + 2 x^{2} )( 1 - 2 x + x^{2} + 2 x^{4} - 8 x^{5} + 8 x^{6} )$
	$1 - 4 x + 7 x^{2} - 6 x^{3} + 4 x^{4} - 12 x^{5} + 28 x^{6} - 32 x^{7} + 16 x^{8}$
Frobenius angles:	$\pm0.0693533547550$, $\pm0.250000000000$, $\pm0.339907131295$, $\pm0.770553776540$
Angle rank:	$2$ (numerical)
Jacobians:	$0$
Isomorphism classes:	3

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2$	$220$	$5018$	$149600$	$686422$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-1$	$3$	$11$	$31$	$19$	$51$	$111$	$223$	$587$	$1043$

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 $\times$ 3.2.ac_b_a 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 : $\Q(\sqrt{-1})$.
• 3.2.ac_b_a : 6.0.2580992.1.

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

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

• 1.16.i : the quaternion algebra over $\Q$ ramified at $2$ and $\infty$.
• 3.16.g_r_ce : 6.0.2580992.1.

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 $\times$ 3.4.ac_f_am. The endomorphism algebra for each factor is:

  • 1.4.a : $\Q(\sqrt{-1})$.
  • 3.4.ac_f_am : 6.0.2580992.1.

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
4.2.a_ab_ac_e	$2$	4.4.ac_j_au_bo
4.2.a_ab_c_e	$2$	4.4.ac_j_au_bo
4.2.e_h_g_e	$2$	4.4.ac_j_au_bo