Abelian variety isogeny class 2.31.a_aci over $\F_{31}$

• Label: 2.31.a_aci
• Base field: $\F_{31}$
• 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:	$\F_{31}$
Dimension:	$2$
L-polynomial:	$1 - 60 x^{2} + 961 x^{4}$
Frobenius angles:	$\pm0.0405348627834$, $\pm0.959465137217$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-2}, \sqrt{-61})\)
Galois group:	$C_2^2$
Jacobians:	$0$
Isomorphism classes:	6

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$902$	$813604$	$887460662$	$849796360336$	$819628270204502$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$32$	$842$	$29792$	$920166$	$28629152$	$887417642$	$27512614112$	$852889100158$	$26439622160672$	$819628253428202$

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 $\F_{31^{2}}$.

Endomorphism algebra over $\F_{31}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-2}, \sqrt{-61})\).

Endomorphism algebra over $\overline{\F}_{31}$

The base change of $A$ to $\F_{31^{2}}$ is 1.961.aci^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-61}) \)$)$

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
2.31.a_ci	$4$	(not in LMFDB)
2.31.ac_c	$8$	(not in LMFDB)
2.31.c_c	$8$	(not in LMFDB)