Abelian variety isogeny class 2.61.ax_je over $\F_{61}$

• Label: 2.61.ax_je
• Base field: $\F_{61}$
• 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:	$\F_{61}$
Dimension:	$2$
L-polynomial:	$1 - 23 x + 238 x^{2} - 1403 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.0340660663953$, $\pm0.341309165886$
Angle rank:	$2$ (numerical)
Number field:	4.0.2213900.2
Galois group:	$D_{4}$
Jacobians:	$4$
Isomorphism classes:	8

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2534$	$13648124$	$51530410400$	$191655910377344$	$713271020537225934$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$39$	$3669$	$227028$	$13842129$	$844511179$	$51519561978$	$3142739029959$	$191707313193729$	$11694146180163348$	$713342911307674429$

Jacobians and polarizations

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

• $y^2=15x^6+12x^5+51x^4+34x^3+16x^2+60$
• $y^2=7x^6+16x^5+39x^4+34x^3+29x^2+51x+31$
• $y^2=21x^6+26x^5+25x^4+50x^3+47x^2+25x+39$
• $y^2=2x^6+45x^5+55x^4+x^3+44x^2+29x+25$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{61}$.

Endomorphism algebra over $\F_{61}$

The endomorphism algebra of this simple isogeny class is 4.0.2213900.2.

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
2.61.x_je	$2$	(not in LMFDB)