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

• Label: 2.31.aq_eu
• Base field: $\F_{31}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• 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:	$1 - 16 x + 124 x^{2} - 496 x^{3} + 961 x^{4}$
Frobenius angles:	$\pm0.179349407864$, $\pm0.298567125522$
Angle rank:	$2$ (numerical)
Number field:	4.0.182528.1
Galois group:	$D_{4}$
Jacobians:	$8$

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$574$	$917252$	$898540174$	$855469574288$	$819925631796334$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$16$	$954$	$30160$	$926310$	$28639536$	$887514330$	$27512526064$	$852890779518$	$26439624254224$	$819628298555674$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):

• $y^2=11 x^6+2 x^5+15 x^4+30 x^3+27 x^2+10 x+15$
• $y^2=2 x^6+23 x^5+28 x^4+20 x^3+18 x^2+5 x+8$
• $y^2=16 x^5+25 x^4+21 x^3+25 x^2+8 x+9$
• $y^2=26 x^6+13 x^5+23 x^4+24 x^2+16 x+18$
• $y^2=29 x^6+12 x^4+16 x^3+2 x^2+12 x+21$
• $y^2=21 x^6+14 x^5+18 x^4+2 x^3+30 x^2+28 x+14$
• $y^2=23 x^6+20 x^5+23 x^4+22 x^3+7 x^2+13 x+29$
• $y^2=24 x^6+26 x^5+10 x^4+22 x^3+6 x^2+22 x+29$

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 4.0.182528.1.

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_eu	$2$	(not in LMFDB)