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

• Label: 2.61.ay_jz
• 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:	$1 - 24 x + 259 x^{2} - 1464 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.113044267146$, $\pm0.295626782064$
Angle rank:	$2$ (numerical)
Number field:	4.0.44688.2
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2493$	$13634217$	$51618472452$	$191778473661273$	$713356271534370573$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$38$	$3664$	$227414$	$13850980$	$844612118$	$51520271470$	$3142742184830$	$191707328053828$	$11694146413364750$	$713342914885487824$

Jacobians and polarizations

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

• $y^2=8x^6+34x^5+15x^4+57x^3+40x^2+5x+54$
• $y^2=47x^6+19x^5+34x^4+36x^3+55x^2+6x+41$
• $y^2=47x^6+46x^5+42x^4+55x^3+23x^2+45x+32$
• $y^2=54x^6+37x^5+47x^4+4x^3+9x^2+53x+15$
• $y^2=9x^6+47x^5+52x^4+54x^3+19x^2+20x+14$
• $y^2=25x^6+16x^5+37x^3+23x^2+34x+1$
• $y^2=37x^6+24x^5+31x^4+54x^3+8x^2+48x+39$
• $y^2=8x^6+49x^5+28x^4+58x^3+6x^2+32x+6$
• $y^2=13x^6+35x^5+36x^4+19x^3+42x^2+37x+29$
• $y^2=37x^6+7x^5+49x^4+3x^3+45x^2+10x+55$
• $y^2=50x^6+26x^5+48x^4+11x^3+55x^2+8x+38$
• $y^2=56x^6+10x^5+25x^4+3x^3+53x^2+50x+46$
• $y^2=13x^6+24x^5+19x^4+43x^3+2x^2+53x+51$
• $y^2=17x^6+17x^5+2x^4+26x^3+44x^2+22x+17$
• $y^2=31x^6+47x^5+4x^4+28x^3+28x^2+38x+44$
• $y^2=35x^6+x^5+16x^4+5x^3+40x^2+5x+30$
• $y^2=51x^6+12x^5+50x^4+26x^3+49x^2+27x+53$
• $y^2=19x^6+x^5+20x^4+14x^2+19x+6$

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 4.0.44688.2.

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.y_jz	$2$	(not in LMFDB)