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

• Label: 2.31.aq_eo
• 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:	$1 - 16 x + 118 x^{2} - 496 x^{3} + 961 x^{4}$
Frobenius angles:	$\pm0.0749296889579$, $\pm0.346262479831$
Angle rank:	$2$ (numerical)
Number field:	4.0.10496.2
Galois group:	$D_{4}$
Jacobians:	$14$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$568$	$904256$	$889856632$	$852474684416$	$819307268864248$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$16$	$942$	$29872$	$923070$	$28617936$	$887436462$	$27512561008$	$852892990974$	$26439637120912$	$819628334086702$

Jacobians and polarizations

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

• $y^2=20 x^6+12 x^5+11 x^4+27 x^3+30 x^2+3 x+11$
• $y^2=15 x^6+5 x^5+27 x^4+11 x^3+18 x^2+15 x+29$
• $y^2=21 x^6+26 x^5+7 x^4+14 x^3+28 x^2+7 x+3$
• $y^2=28 x^6+3 x^5+10 x^4+17 x^3+13 x^2+6 x+28$
• $y^2=12 x^5+18 x^4+11 x^3+4 x^2+3 x+17$
• $y^2=24 x^6+16 x^5+x^4+17 x^2+17 x+17$
• $y^2=21 x^6+17 x^5+20 x^4+16 x^3+24 x^2+9 x+11$
• $y^2=12 x^6+4 x^5+11 x^4+30 x^3+9 x^2+17 x+3$
• $y^2=12 x^6+28 x^5+14 x^4+4 x^3+13 x^2+26 x+22$
• $y^2=3 x^6+11 x^5+19 x^3+5 x^2+7 x+13$
• $y^2=11 x^6+7 x^5+2 x^4+9 x^3+11 x^2+3 x+24$
• $y^2=16 x^6+14 x^5+26 x^4+29 x^3+25 x^2+18 x+30$
• $y^2=29 x^6+6 x^5+21 x^4+15 x^3+25 x^2+7 x+21$
• $y^2=21 x^6+27 x^5+4 x^4+13 x^3+5 x^2+28 x+10$

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 4.0.10496.2.

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.q_eo	$2$	(not in LMFDB)