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

• Label: 2.61.ay_jw
• 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:	$1 - 24 x + 256 x^{2} - 1464 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.0772899280240$, $\pm0.308576229556$
Angle rank:	$2$ (numerical)
Number field:	4.0.3744000.2
Galois group:	$D_{4}$
Jacobians:	$16$
Isomorphism classes:	32
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2490$	$13610340$	$51569197290$	$191725538657040$	$713319176976056250$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$38$	$3658$	$227198$	$13847158$	$844568198$	$51519917338$	$3142740184958$	$191707320328798$	$11694146382954758$	$713342914568337898$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 16 curves (of which all are hyperelliptic):

• $y^2=38 x^6+10 x^5+11 x^4+39 x^3+39 x^2+3 x+53$
• $y^2=33 x^6+53 x^5+49 x^4+5 x^3+5 x^2+41 x+54$
• $y^2=53 x^6+59 x^5+58 x^4+50 x^3+11 x^2+52 x+32$
• $y^2=52 x^6+15 x^4+54 x^3+34 x^2+58 x+23$
• $y^2=6 x^6+15 x^5+12 x^4+17 x^3+7 x^2+32 x+22$
• $y^2=4 x^6+24 x^5+25 x^4+28 x^3+47 x^2+59 x+28$
• $y^2=23 x^6+12 x^5+34 x^4+40 x^3+52 x^2+56 x+9$
• $y^2=9 x^6+37 x^5+31 x^4+31 x^3+46 x^2+2 x+24$
• $y^2=29 x^6+42 x^5+59 x^4+27 x^3+17 x^2+33 x+2$
• $y^2=51 x^6+56 x^5+17 x^4+55 x^3+20 x^2+18 x+12$
• $y^2=8 x^6+4 x^5+9 x^4+13 x^3+21 x^2+53 x+57$
• $y^2=55 x^6+49 x^5+30 x^4+28 x^3+7 x^2+45 x+37$
• $y^2=29 x^6+14 x^5+57 x^4+42 x^3+24 x^2+5 x+42$
• $y^2=7 x^6+32 x^5+43 x^4+29 x^3+59 x^2+60 x+45$
• $y^2=45 x^6+5 x^5+29 x^4+20 x^3+50 x^2+19 x+55$
• $y^2=55 x^6+17 x^5+51 x^4+7 x^3+32 x^2+2 x+49$

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 4.0.3744000.2.

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_jw	$2$	(not in LMFDB)