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

• Label: 2.31.aq_er
• 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:	$1 - 16 x + 121 x^{2} - 496 x^{3} + 961 x^{4}$
Frobenius angles:	$\pm0.128815581612$, $\pm0.326819455035$
Angle rank:	$2$ (numerical)
Number field:	4.0.43625.1
Galois group:	$D_{4}$
Jacobians:	$12$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$571$	$910745$	$894195136$	$853988282345$	$819637027021291$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$16$	$948$	$30016$	$924708$	$28629456$	$887489598$	$27512724976$	$852893526468$	$26439641292736$	$819628360444948$

Jacobians and polarizations

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

• $y^2=3 x^6+15 x^5+30 x^4+6 x^3+28 x^2+24 x+13$
• $y^2=27 x^6+6 x^5+5 x^4+8 x^3+9 x^2+27 x+6$
• $y^2=21 x^6+24 x^5+11 x^4+26 x^3+27 x^2+6 x+3$
• $y^2=30 x^6+3 x^5+24 x^4+19 x^3+17 x^2+22 x+26$
• $y^2=21 x^6+8 x^5+22 x^4+7 x^3+26 x^2+5 x+8$
• $y^2=24 x^6+28 x^5+14 x^4+25 x^3+11 x^2+23 x+23$
• $y^2=26 x^6+19 x^5+7 x^4+15 x^3+4 x^2+27 x+12$
• $y^2=21 x^6+17 x^5+10 x^4+13 x^3+27 x^2+30 x+12$
• $y^2=13 x^6+22 x^5+16 x^4+10 x^3+8 x^2+26 x+10$
• $y^2=18 x^6+10 x^5+30 x^4+17 x^3+20 x^2+6 x+22$
• $y^2=30 x^6+6 x^5+x^4+8 x^3+11 x^2+7 x+22$
• $y^2=12 x^6+12 x^5+3 x^4+30 x^3+2 x^2+27 x+29$

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 4.0.43625.1.

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_er	$2$	(not in LMFDB)