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

• Label: 2.31.d_bd
• 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:	$1 + 3 x + 29 x^{2} + 93 x^{3} + 961 x^{4}$
Frobenius angles:	$\pm0.369541398590$, $\pm0.732797074295$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\sqrt{-346 -6 \sqrt{141}})\)
Galois group:	$D_{4}$
Jacobians:	$18$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1087$	$972865$	$888812725$	$854748487485$	$819205484491312$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$35$	$1011$	$29837$	$925531$	$28614380$	$887438463$	$27513008075$	$852891190051$	$26439630539177$	$819628283456526$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 18 curves (of which all are hyperelliptic):

• $y^2=21 x^6+26 x^5+22 x^4+19 x^2+10 x+3$
• $y^2=9 x^6+8 x^5+23 x^4+5 x^3+22 x^2+12 x+29$
• $y^2=x^6+16 x^5+22 x^3+18 x^2+16 x+14$
• $y^2=16 x^6+18 x^5+11 x^4+19 x^3+10 x^2+13 x+27$
• $y^2=30 x^6+21 x^5+25 x^4+11 x^3+17 x^2+3 x+24$
• $y^2=16 x^6+18 x^5+5 x^4+26 x^3+22 x^2+13 x+1$
• $y^2=16 x^6+22 x^5+10 x^4+17 x^3+18 x^2+25 x+5$
• $y^2=10 x^6+16 x^5+28 x^4+4 x^3+18 x^2+4 x+24$
• $y^2=19 x^6+15 x^5+11 x^4+30 x^3+19 x^2+23$
• $y^2=20 x^6+16 x^5+13 x^4+27 x^3+8 x^2+18 x+10$
• $y^2=16 x^6+25 x^5+12 x^4+3 x^3+14 x^2+26 x+8$
• $y^2=25 x^6+2 x^5+16 x^4+7 x^3+19 x^2+30 x+27$
• $y^2=24 x^6+9 x^5+15 x^4+25 x^3+30 x^2+4 x+18$
• $y^2=6 x^6+18 x^5+4 x^4+25 x^3+5 x^2+5 x+18$
• $y^2=9 x^6+27 x^5+30 x^4+23 x^3+20 x^2+25 x+24$
• $y^2=11 x^6+24 x^5+17 x^4+23 x^3+10 x^2+30 x+7$
• $y^2=4 x^6+x^5+14 x^4+23 x^3+9 x+29$
• $y^2=4 x^5+30 x^4+6 x^3+21 x^2+13 x+29$

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 \(\Q(\sqrt{-346 -6 \sqrt{141}})\).

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.ad_bd	$2$	(not in LMFDB)