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

• Label: 2.31.l_dg
• 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:	$1 + 11 x + 84 x^{2} + 341 x^{3} + 961 x^{4}$
Frobenius angles:	$\pm0.575828709429$, $\pm0.770839730243$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\sqrt{-342 -22 \sqrt{33}})\)
Galois group:	$D_{4}$
Jacobians:	$12$
Isomorphism classes:	12
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1398$	$970212$	$875153592$	$853576994208$	$819638011382898$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$43$	$1009$	$29374$	$924265$	$28629493$	$887534890$	$27512390515$	$852890045233$	$26439640887490$	$819628210587049$

Jacobians and polarizations

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

• $y^2=10 x^6+26 x^5+21 x^3+5 x^2+5 x+13$
• $y^2=10 x^6+25 x^5+8 x^4+9 x^2+23 x+22$
• $y^2=19 x^6+28 x^5+26 x^4+7 x^3+4 x^2+15 x+10$
• $y^2=16 x^6+23 x^5+8 x^4+17 x^3+29 x^2+13 x+20$
• $y^2=18 x^6+2 x^5+22 x^4+17 x^3+25 x^2+23 x+15$
• $y^2=26 x^6+4 x^5+25 x^4+15 x^3+18 x^2+14 x+8$
• $y^2=9 x^6+2 x^5+3 x^4+19 x^3+3 x^2+26 x+14$
• $y^2=2 x^6+18 x^5+28 x^4+28 x^3+8 x^2+29 x+10$
• $y^2=12 x^6+8 x^5+9 x^4+17 x^2+17 x+27$
• $y^2=18 x^6+23 x^5+7 x^4+29 x^3+25 x^2+15 x+15$
• $y^2=19 x^6+17 x^5+27 x^4+11 x^3+x^2+8 x+28$
• $y^2=17 x^6+17 x^5+3 x^4+6 x^3+4 x^2+25 x+18$

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 \(\Q(\sqrt{-342 -22 \sqrt{33}})\).

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.al_dg	$2$	(not in LMFDB)