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

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

Invariants

Base field:	$\F_{31}$
Dimension:	$2$
L-polynomial:	$1 + 6 x + 68 x^{2} + 186 x^{3} + 961 x^{4}$
Frobenius angles:	$\pm0.536323198846$, $\pm0.639708574347$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\sqrt{-112 -6 \sqrt{3}})\)
Galois group:	$D_{4}$
Jacobians:	$14$
Isomorphism classes:	14
Cyclic group of points:	yes

This isogeny class is simple and geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1222$	$1024036$	$874195582$	$851625202896$	$820042543756462$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$38$	$1062$	$29342$	$922150$	$28643618$	$887497782$	$27512353466$	$852891630334$	$26439623522822$	$819628292641302$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 14 curves (of which all are hyperelliptic):

• $y^2=2 x^6+5 x^5+5 x^4+27 x^3+4 x^2+18 x+6$
• $y^2=26 x^6+21 x^5+9 x^4+21 x^3+3 x^2+21 x+20$
• $y^2=18 x^6+x^5+11 x^4+27 x^3+5 x^2+22 x+4$
• $y^2=9 x^6+15 x^5+15 x^4+15 x^3+7 x^2+x+2$
• $y^2=x^5+8 x^4+21 x^3+5 x^2+10 x+9$
• $y^2=21 x^6+26 x^5+14 x^4+2 x^3+11 x^2+5 x+6$
• $y^2=5 x^6+x^5+15 x^4+12 x^3+27 x^2+6 x+24$
• $y^2=14 x^6+29 x^4+9 x^3+8 x^2+12 x$
• $y^2=28 x^6+24 x^5+24 x^4+23 x^3+4 x^2+20 x+3$
• $y^2=x^6+8 x^5+22 x^4+28 x^3+4 x^2+14 x+19$
• $y^2=4 x^6+11 x^5+10 x^4+17 x^3+10 x^2+29 x+6$
• $y^2=23 x^6+14 x^5+13 x^4+26 x^3+7 x^2+25 x+29$
• $y^2=19 x^6+23 x^5+18 x^4+8 x^3+8 x^2+30 x+3$
• $y^2=25 x^6+11 x^5+24 x^4+3 x^3+17 x^2+25 x+26$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{31}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-112 -6 \sqrt{3}})\).

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.ag_cq	$2$	(not in LMFDB)