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

• Label: 2.61.ax_jl
• 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:	$1 - 23 x + 245 x^{2} - 1403 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.119010327655$, $\pm0.317853749233$
Angle rank:	$2$ (numerical)
Number field:	4.0.7684197.1
Galois group:	$D_{4}$
Jacobians:	$18$
Isomorphism classes:	18

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2541$	$13703613$	$51640511769$	$191767387365477$	$713343767808435456$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$39$	$3683$	$227511$	$13850179$	$844597314$	$51520222379$	$3142743208875$	$191707344090595$	$11694146499633555$	$713342914490048918$

Jacobians and polarizations

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

• $y^2=32x^6+58x^5+55x^4+19x^3+25x^2+6x+21$
• $y^2=37x^6+27x^5+22x^4+4x^3+43x^2+29x+3$
• $y^2=21x^6+51x^5+11x^4+60x^3+35x^2+42x+31$
• $y^2=10x^6+23x^5+23x^4+5x^3+57x^2+26x+50$
• $y^2=26x^6+27x^5+38x^3+2x^2+60x+40$
• $y^2=8x^6+54x^5+35x^4+10x^3+50x^2+26x+33$
• $y^2=40x^6+54x^5+6x^4+36x^3+13x^2+54x+13$
• $y^2=6x^6+43x^5+28x^4+46x^3+50x^2+42x+47$
• $y^2=54x^6+13x^5+36x^3+51x^2+33x+6$
• $y^2=16x^6+28x^5+43x^4+54x^3+30x^2+26x+10$
• $y^2=51x^6+35x^5+19x^4+2x^3+19x^2+51x+53$
• $y^2=4x^6+24x^5+58x^4+17x^3+29x^2+48x+3$
• $y^2=33x^6+22x^5+52x^4+30x^3+3x^2+42x+7$
• $y^2=34x^6+36x^5+25x^4+59x^3+46x^2+42x+52$
• $y^2=51x^6+57x^5+41x^4+7x^3+57x^2+36x+44$
• $y^2=60x^6+38x^5+60x^4+43x^3+3x^2+4x+55$
• $y^2=58x^6+38x^5+27x^4+12x^3+38x^2+29x+40$
• $y^2=7x^6+36x^5+51x^4+33x^3+48x^2+52x+31$

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 4.0.7684197.1.

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
2.61.x_jl	$2$	(not in LMFDB)