Abelian variety isogeny class 2.2.a_d over $\F_{2}$

• Label: 2.2.a_d
• Base field: $\F_{2}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{2}$
Dimension:	$2$
L-polynomial:	$( 1 - x + 2 x^{2} )( 1 + x + 2 x^{2} )$
	$1 + 3 x^{2} + 4 x^{4}$
Frobenius angles:	$\pm0.384973271919$, $\pm0.615026728081$
Angle rank:	$1$ (numerical)
Jacobians:	$0$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$8$	$64$	$56$	$256$	$968$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$3$	$11$	$9$	$15$	$33$	$47$	$129$	$319$	$513$	$911$

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 $\F_{2^{2}}$.

Endomorphism algebra over $\F_{2}$

The isogeny class factors as 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.ab : \(\Q(\sqrt{-7}) \).
• 1.2.b : \(\Q(\sqrt{-7}) \).

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

The base change of $A$ to $\F_{2^{2}}$ is 1.4.d^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$

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
2.2.ac_f	$2$	2.4.g_r
2.2.c_f	$2$	2.4.g_r
2.2.a_ad	$4$	2.16.ac_bh