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

• Label: 2.61.ax_jj
• 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:	$1 - 23 x + 243 x^{2} - 1403 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.100123618576$, $\pm0.325377247674$
Angle rank:	$2$ (numerical)
Number field:	4.0.103725.1
Galois group:	$D_{4}$
Jacobians:	$24$
Isomorphism classes:	30

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2539$	$13687749$	$51609046975$	$191735811639909$	$713323953851741584$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$39$	$3679$	$227373$	$13847899$	$844573854$	$51520066783$	$3142742647629$	$191707344257299$	$11694146509589703$	$713342914532262454$

Jacobians and polarizations

This isogeny class contains the Jacobians of 24 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=17x^6+15x^5+4x^4+8x^3+28x^2+13x+28$
• $y^2=46x^6+48x^5+35x^4+27x^3+55x^2+20x+13$
• $y^2=21x^6+9x^5+33x^4+43x^3+3x^2+24x+8$
• $y^2=31x^6+23x^5+38x^4+49x^3+34x^2+14x+10$
• $y^2=26x^6+19x^5+6x^4+27x^3+49x^2+19x+47$
• $y^2=40x^6+56x^5+40x^4+17x^3+6x^2+27x+31$
• $y^2=44x^6+33x^5+35x^4+46x^3+34x^2+33x+17$
• $y^2=23x^6+39x^5+28x^4+2x^3+30x^2+36x+33$
• $y^2=33x^6+33x^5+59x^4+19x^3+34x^2+38x+8$
• $y^2=55x^6+22x^5+36x^4+54x^3+20x^2+35x+18$
• $y^2=17x^6+36x^5+14x^4+11x^3+57x^2+43x+55$
• $y^2=2x^6+20x^5+48x^4+19x^3+53x^2+7x+10$
• $y^2=53x^6+24x^5+33x^4+47x^3+43x^2+59x+39$
• $y^2=54x^6+22x^5+53x^4+57x^3+18x^2+16x+55$
• $y^2=56x^6+13x^5+9x^4+41x^3+x^2+2x+27$
• $y^2=52x^6+52x^5+25x^4+43x^3+6x^2+47x+6$
• $y^2=33x^6+58x^5+7x^4+14x^3+17x^2+28x+54$
• $y^2=58x^6+53x^5+19x^4+27x^3+x^2+53x+17$
• $y^2=29x^6+30x^5+7x^4+x^3+12x^2+21x$
• $y^2=25x^6+41x^5+16x^4+41x^3+54x^2+3x+53$
• $y^2=48x^6+5x^5+19x^4+9x^3+19x^2+8x$
• $y^2=58x^6+41x^5+38x^4+32x^3+5x^2+44x+26$
• $y^2=29x^6+54x^5+33x^4+51x^3+35x^2+x+15$
• $y^2=35x^6+19x^5+15x^4+37x^3+7x^2+26x+21$

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 4.0.103725.1.

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_jj	$2$	(not in LMFDB)