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

• Label: 2.61.ax_jk
• Base field: $\F_{61}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• 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:	$1 - 23 x + 244 x^{2} - 1403 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.109726878648$, $\pm0.321721348102$
Angle rank:	$2$ (numerical)
Number field:	4.0.512705.1
Galois group:	$D_{4}$
Jacobians:	$20$
Isomorphism classes:	20

This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

$p$-rank:	$1$
Slopes:	$[0, 1/2, 1/2, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2540$	$13695680$	$51624778640$	$191751626981120$	$713333957925433500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$39$	$3681$	$227442$	$13849041$	$844585699$	$51520147878$	$3142742990559$	$191707345034241$	$11694146513673882$	$713342914584741001$

Jacobians and polarizations

This isogeny class contains the Jacobians of 20 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=48x^6+29x^5+60x^4+33x^3+27x^2+51x+23$
• $y^2=21x^6+18x^5+34x^3+20x^2+6x+52$
• $y^2=7x^6+11x^5+11x^4+22x^3+34x^2+20x+45$
• $y^2=16x^6+20x^5+30x^4+8x^3+54x^2+52x+33$
• $y^2=23x^6+42x^5+37x^4+31x^3+23x^2+11x+9$
• $y^2=32x^6+50x^4+26x^3+58x^2+32x+51$
• $y^2=17x^6+31x^5+54x^4+34x^3+34x^2+32x+11$
• $y^2=48x^6+27x^5+5x^4+52x^3+42x^2+55x+32$
• $y^2=46x^6+14x^5+21x^4+16x^3+50x^2+11x+54$
• $y^2=44x^6+59x^5+52x^4+13x^3+51x^2+9x+32$
• $y^2=10x^6+45x^5+38x^4+14x^3+2x^2+46x+52$
• $y^2=31x^6+30x^5+5x^4+27x^3+9x^2+49x+50$
• $y^2=42x^6+45x^5+47x^4+42x^3+49x^2+15x+59$
• $y^2=6x^6+15x^5+36x^4+55x^3+39x^2+15x+25$
• $y^2=21x^6+23x^5+27x^4+51x^3+43x^2+37x+50$
• $y^2=43x^5+34x^4+44x^3+40x^2+37x+13$
• $y^2=17x^6+41x^5+39x^4+29x^3+29x^2+16x+12$
• $y^2=46x^6+36x^5+7x^4+8x^3+13x^2+19x+41$
• $y^2=30x^6+5x^5+17x^4+58x^3+23x^2+39x+33$
• $y^2=33x^6+33x^5+9x^4+28x^3+20x^2+8x+58$

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 4.0.512705.1.

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_jk	$2$	(not in LMFDB)