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

• Label: 2.31.j_dd
• 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:	$1 + 9 x + 81 x^{2} + 279 x^{3} + 961 x^{4}$
Frobenius angles:	$\pm0.598225233254$, $\pm0.668328438074$
Angle rank:	$2$ (numerical)
Number field:	4.0.260125.1
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1331$	$1004905$	$869157641$	$853225644205$	$820080853986416$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$41$	$1043$	$29171$	$923883$	$28644956$	$887427623$	$27512514581$	$852893337763$	$26439615552611$	$819628260989198$

Jacobians and polarizations

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

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

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 4.0.260125.1.

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.aj_dd	$2$	(not in LMFDB)