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

• Label: 2.31.u_gg
• Base field: $\F_{31}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{31}$
Dimension:	$2$
L-polynomial:	$( 1 + 10 x + 31 x^{2} )^{2}$
	$1 + 20 x + 162 x^{2} + 620 x^{3} + 961 x^{4}$
Frobenius angles:	$\pm0.854999228987$, $\pm0.854999228987$
Angle rank:	$1$ (numerical)
Jacobians:	$3$

This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1764$	$853776$	$891739044$	$853776000000$	$819230447328804$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$52$	$886$	$29932$	$924478$	$28615252$	$887613046$	$27511951372$	$852894274558$	$26439610334452$	$819628304892406$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 3 curves (of which all are hyperelliptic):

• $y^2=26 x^6+2 x^4+2 x^2+26$
• $y^2=x^6+x^3+16$
• $y^2=4 x^6+25 x^5+5 x^4+27 x^3+5 x^2+25 x+4$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{31}$.

Endomorphism algebra over $\F_{31}$

The isogeny class factors as 1.31.k^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-6}) \)$)$

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
2.31.au_gg	$2$	(not in LMFDB)
2.31.a_abm	$2$	(not in LMFDB)
2.31.ak_cr	$3$	(not in LMFDB)