Abelian variety isogeny class 2.4.a_ah over $\F_{2^{2}}$

• Label: 2.4.a_ah
• Base field: $\F_{2^{2}}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{2^{2}}$
Dimension:	$2$
L-polynomial:	$1 - 7 x^{2} + 16 x^{4}$
Frobenius angles:	$\pm0.0804306232552$, $\pm0.919569376745$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(i, \sqrt{15})\)
Galois group:	$C_2^2$
Jacobians:	$0$
Isomorphism classes:	2

This isogeny class is simple but not geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10$	$100$	$4090$	$57600$	$1050250$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$5$	$3$	$65$	$223$	$1025$	$4083$	$16385$	$65983$	$262145$	$1051923$

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 $\F_{2^{2}}$

The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{15})\).

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

The base change of $A$ to $\F_{2^{4}}$ is 1.16.ah^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$

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
2.4.ac_j	$4$	2.256.abi_bev
2.4.a_h	$4$	2.256.abi_bev
2.4.c_j	$4$	2.256.abi_bev