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

• Label: 2.61.t_hz
• 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:	$1 + 19 x + 207 x^{2} + 1159 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.652685161311$, $\pm0.772295141995$
Angle rank:	$2$ (numerical)
Number field:	4.0.8889237.1
Galois group:	$D_{4}$
Jacobians:	$32$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5107$	$14049357$	$51189243343$	$191841511197525$	$713331613958991952$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$81$	$3775$	$225519$	$13855531$	$844582926$	$51520121287$	$3142744627551$	$191707310794771$	$11694146106921369$	$713342910716455030$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 32 curves (of which all are hyperelliptic):

• $y^2=33 x^6+22 x^5+22 x^4+20 x^3+45 x^2+24 x+42$
• $y^2=6 x^6+48 x^5+33 x^4+32 x^3+15 x^2+45 x+44$
• $y^2=60 x^6+55 x^5+52 x^4+31 x^3+24 x^2+5 x+10$
• $y^2=3 x^6+40 x^5+12 x^4+45 x^3+32 x^2+44 x+8$
• $y^2=3 x^6+20 x^5+46 x^4+21 x^3+45 x^2+8 x+43$
• $y^2=14 x^6+54 x^5+16 x^4+42 x^3+14 x^2+24 x+24$
• $y^2=10 x^6+34 x^5+49 x^4+7 x^3+28 x^2+34 x+4$
• $y^2=58 x^6+55 x^5+6 x^4+55 x^3+5 x^2+6 x+32$
• $y^2=14 x^6+24 x^5+56 x^3+9 x^2+51$
• $y^2=34 x^6+21 x^5+33 x^4+49 x^3+38 x^2+7 x+11$
• $y^2=11 x^6+19 x^4+4 x^3+13 x^2+27 x+3$
• $y^2=6 x^6+33 x^5+10 x^3+12 x^2+35 x+55$
• $y^2=4 x^6+37 x^5+59 x^4+20 x^3+52 x^2+15 x+9$
• $y^2=15 x^6+15 x^5+28 x^4+16 x^3+23 x^2+11 x+41$
• $y^2=52 x^6+3 x^5+49 x^4+12 x^3+59 x^2+22 x+5$
• $y^2=9 x^6+44 x^5+54 x^4+17 x^3+17 x^2+8 x+36$
• $y^2=49 x^6+11 x^5+36 x^4+51 x^3+51 x^2+13 x+22$
• $y^2=25 x^6+38 x^5+40 x^4+15 x^3+20 x+48$
• $y^2=5 x^6+59 x^5+41 x^4+26 x^3+3 x^2+23 x+56$
• $y^2=18 x^6+7 x^5+57 x^4+44 x^3+3 x^2+59 x+42$
• and

• $y^2=40 x^6+57 x^4+34 x^3+18 x^2+54 x+22$
• $y^2=32 x^6+7 x^5+51 x^4+3 x^3+20 x^2+54 x+13$
• $y^2=8 x^6+27 x^5+9 x^4+28 x^3+37 x^2+30 x+22$
• $y^2=49 x^6+2 x^5+12 x^4+6 x^3+12 x^2+8 x+8$
• $y^2=5 x^6+49 x^5+4 x^4+50 x^3+12 x^2+53 x+10$
• $y^2=51 x^6+46 x^5+56 x^4+38 x^3+2 x^2+49 x+39$
• $y^2=35 x^6+40 x^5+45 x^4+2 x^3+16 x^2+7 x+19$
• $y^2=24 x^6+23 x^5+19 x^4+57 x^3+29 x^2+35 x+11$
• $y^2=7 x^6+48 x^5+23 x^4+8 x^3+x^2+36 x+12$
• $y^2=56 x^6+60 x^5+49 x^4+58 x^3+41 x^2+5 x+16$
• $y^2=43 x^6+40 x^5+35 x^4+16 x^3+16 x^2+28 x+50$
• $y^2=46 x^6+30 x^5+51 x^4+27 x^3+51 x^2+49 x+42$

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 4.0.8889237.1.

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.at_hz	$2$	(not in LMFDB)