Abelian variety isogeny class 2.2.a_ac over F2\F_{2}

• Label: 2.2.a_ac
• Base field: $\F_{2}$
• Dimension: $2$
• pp-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:	$1 - 2 x^{2} + 4 x^{4}$
Frobenius angles:	$\pm0.166666666667$, $\pm0.833333333333$
Angle rank:	$0$ (numerical)
Number field:	$\Q(\sqrt{-2}, \sqrt{-3})$
Galois group:	$C_2^2$
Jacobians:	$0$
Isomorphism classes:	2

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3$	$9$	$81$	$441$	$993$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$3$	$1$	$9$	$25$	$33$	$97$	$129$	$289$	$513$	$961$

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 $\F_{2^{6}}$.

Endomorphism algebra over $\F_{2}$

The endomorphism algebra of this simple isogeny class is $\Q(\sqrt{-2}, \sqrt{-3})$.

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

The base change of $A$ to $\F_{2^{6}}$ is 1.64.q^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is 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.ac^2 and its endomorphism algebra is $\mathrm{M}_{2}($$\Q(\sqrt{-3})$$)$

• Endomorphism algebra over $\F_{2^{3}}$

  The base change of $A$ to $\F_{2^{3}}$ is 1.8.a^2 and its endomorphism algebra is $\mathrm{M}_{2}($$\Q(\sqrt{-2})$$)$

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_e	$3$	2.8.a_q
2.2.a_c	$4$	2.16.i_bw
2.2.ac_c	$8$	2.256.bg_bdo