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

• Label: 2.31.an_do
• 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:	$( 1 - 10 x + 31 x^{2} )( 1 - 3 x + 31 x^{2} )$
	$1 - 13 x + 92 x^{2} - 403 x^{3} + 961 x^{4}$
Frobenius angles:	$\pm0.145000771013$, $\pm0.413172001920$
Angle rank:	$2$ (numerical)
Jacobians:	$12$
Isomorphism classes:	100
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$638$	$937860$	$892967768$	$852514740000$	$819527410949258$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$19$	$977$	$29974$	$923113$	$28625629$	$887554442$	$27513258379$	$852893716273$	$26439621554794$	$819628243511177$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):

• $y^2=24 x^6+x^5+22 x^4+9 x^3+4 x^2+27 x+14$
• $y^2=3 x^6+12 x^5+10 x^4+2 x^3+21 x^2+17$
• $y^2=12 x^6+21 x^5+19 x^4+19 x^3+7 x^2+23 x+22$
• $y^2=23 x^6+15 x^5+4 x^4+x^3+21 x^2+20 x+22$
• $y^2=21 x^6+6 x^5+13 x^4+5 x^3+7 x^2+30 x+12$
• $y^2=19 x^6+26 x^5+15 x^4+21 x^3+7 x^2+2 x+14$
• $y^2=26 x^6+23 x^5+4 x^4+27 x^3+11 x^2+12 x+24$
• $y^2=11 x^6+21 x^5+16 x^3+21 x^2+3 x+29$
• $y^2=22 x^6+29 x^5+16 x^4+13 x^3+14 x^2+2$
• $y^2=22 x^6+4 x^5+25 x^4+25 x^3+25 x^2+26 x+21$
• $y^2=27 x^6+3 x^5+21 x^4+28 x^3+x^2+22 x+25$
• $y^2=16 x^6+4 x^5+10 x^4+30 x^3+13 x+9$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{31}$

The isogeny class factors as 1.31.ak $\times$ 1.31.ad and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.31.ak : \(\Q(\sqrt{-6}) \).
• 1.31.ad : \(\Q(\sqrt{-115}) \).

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.ah_bg	$2$	(not in LMFDB)
2.31.h_bg	$2$	(not in LMFDB)
2.31.n_do	$2$	(not in LMFDB)