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

• Label: 2.31.ar_ez
• 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:	$1 - 17 x + 129 x^{2} - 527 x^{3} + 961 x^{4}$
Frobenius angles:	$\pm0.0793535070825$, $\pm0.311738106719$
Angle rank:	$2$ (numerical)
Number field:	4.0.284445.1
Galois group:	$D_{4}$
Jacobians:	$4$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$547$	$894345$	$890010025$	$853195292205$	$819474555124912$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$15$	$931$	$29877$	$923851$	$28623780$	$887444503$	$27512392755$	$852891761251$	$26439636995637$	$819628386019726$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):

• $y^2=22 x^6+5 x^5+29 x^4+17 x^3+3 x^2+23 x+6$
• $y^2=21 x^6+18 x^5+25 x^4+20 x^3+15 x$
• $y^2=30 x^6+21 x^5+30 x^4+22 x^3+23 x^2+6 x+23$
• $y^2=24 x^6+4 x^5+27 x^4+25 x^3+20 x^2+x+21$

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 4.0.284445.1.

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.r_ez	$2$	(not in LMFDB)