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

• Label: 2.2.a_ae
• Base field: $\F_{2}$
• Dimension: $2$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{2}$
Dimension:	$2$
L-polynomial:	$( 1 - 2 x^{2} )^{2}$
	$1 - 4 x^{2} + 4 x^{4}$
Frobenius angles:	$0$, $0$, $1$, $1$
Angle rank:	$0$ (numerical)
Number field:	\(\Q(\sqrt{2}) \)
Galois group:	$C_2$
Jacobians:	$0$
Cyclic group of points:	yes

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

Newton polygon

This isogeny class is supersingular.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1$	$1$	$49$	$81$	$961$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$3$	$-3$	$9$	$1$	$33$	$33$	$129$	$193$	$513$	$897$

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 endomorphism algebra of this simple isogeny class is the quaternion algebra over \(\Q(\sqrt{2}) \) ramified at both real infinite places.

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

The base change of $A$ to $\F_{2^{2}}$ is 1.4.ae^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.

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.a_c	$3$	2.8.a_aq
2.2.a_e	$4$	2.16.aq_ds
2.2.a_c	$6$	2.64.abg_ou

Additional information

This isogeny class appears as a sporadic example in the classification of abelian varieties with one rational point.