Abelian variety isogeny class 2.8.a_ap over $\F_{2^{3}}$

• Label: 2.8.a_ap
• Base field: $\F_{2^{3}}$
• 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:	$\F_{2^{3}}$
Dimension:	$2$
L-polynomial:	$1 - 15 x^{2} + 64 x^{4}$
Frobenius angles:	$\pm0.0565670411287$, $\pm0.943432958871$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(i, \sqrt{31})\)
Galois group:	$C_2^2$
Jacobians:	$0$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$50$	$2500$	$261650$	$16000000$	$1073755250$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$9$	$35$	$513$	$3903$	$32769$	$261155$	$2097153$	$16774783$	$134217729$	$1073768675$

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 $\F_{2^{3}}$

The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{31})\).

Endomorphism algebra over $\overline{\F}_{2^{3}}$

The base change of $A$ to $\F_{2^{6}}$ is 1.64.ap^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-31}) \)$)$

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
2.8.ac_r	$4$	(not in LMFDB)
2.8.a_p	$4$	(not in LMFDB)
2.8.c_r	$4$	(not in LMFDB)