Abelian variety isogeny class 2.71.a_afk over $\F_{71}$

• Label: 2.71.a_afk
• Base field: $\F_{71}$
• 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:	$\F_{71}$
Dimension:	$2$
L-polynomial:	$1 - 140 x^{2} + 5041 x^{4}$
Frobenius angles:	$\pm0.0267434241185$, $\pm0.973256575882$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-2}, \sqrt{-141})\)
Galois group:	$C_2^2$
Jacobians:	$0$
Isomorphism classes:	8
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4902$	$24029604$	$128099657142$	$645269935882896$	$3255243548601824502$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$72$	$4762$	$357912$	$25392646$	$1804229352$	$128099030362$	$9095120158392$	$645753451707838$	$45848500718449032$	$3255243546193767802$

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 $\F_{71^{2}}$.

Endomorphism algebra over $\F_{71}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-2}, \sqrt{-141})\).

Endomorphism algebra over $\overline{\F}_{71}$

The base change of $A$ to $\F_{71^{2}}$ is 1.5041.afk^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-141}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.71.a_fk	$4$	(not in LMFDB)
2.71.ac_c	$8$	(not in LMFDB)
2.71.c_c	$8$	(not in LMFDB)