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

• Label: 2.61.u_io
• Base field: $\F_{61}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{61}$
Dimension:	$2$
L-polynomial:	$( 1 + 10 x + 61 x^{2} )^{2}$
	$1 + 20 x + 222 x^{2} + 1220 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.721142061624$, $\pm0.721142061624$
Angle rank:	$1$ (numerical)
Jacobians:	$49$

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5184$	$14017536$	$51144727104$	$191900067840000$	$713310903511331904$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$82$	$3766$	$225322$	$13859758$	$844558402$	$51519904486$	$3142749846682$	$191707271553118$	$11694146079625522$	$713342914323062806$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 49 curves (of which all are hyperelliptic):

• $y^2=3 x^6+29 x^5+48 x^3+29 x^2+34 x+41$
• $y^2=16 x^6+42 x^5+23 x^4+50 x^3+37 x^2+47 x+48$
• $y^2=8 x^6+10 x^5+53 x^4+12 x^3+50 x^2+17 x+37$
• $y^2=56 x^6+21 x^5+36 x^4+56 x^3+36 x^2+21 x+56$
• $y^2=54 x^6+43 x^4+43 x^2+54$
• $y^2=43 x^6+45 x^4+45 x^2+43$
• $y^2=48 x^6+20 x^4+20 x^2+48$
• $y^2=57 x^6+36 x^4+36 x^2+57$
• $y^2=56 x^6+12 x^5+32 x^4+55 x^3+38 x^2+56 x+46$
• $y^2=34 x^6+29 x^4+29 x^2+34$
• $y^2=25 x^6+44 x^5+10 x^4+36 x^3+x^2+36 x+36$
• $y^2=34 x^6+45 x^5+33 x^4+4 x^3+31 x^2+14 x+52$
• $y^2=7 x^6+47 x^5+21 x^4+32 x^3+56 x^2+17 x+8$
• $y^2=x^6+19 x^3+27$
• $y^2=20 x^6+12 x^5+44 x^4+36 x^3+43 x^2+x+52$
• $y^2=54 x^6+17 x^5+38 x^4+3 x^3+38 x^2+17 x+54$
• $y^2=3 x^6+3 x^5+49 x^4+16 x^3+39 x^2+56 x+36$
• $y^2=50 x^6+16 x^5+35 x^4+6 x^3+30 x^2+13 x+37$
• $y^2=39 x^6+57 x^5+18 x^4+19 x^3+18 x^2+57 x+39$
• $y^2=x^6+x^3+52$
• and

• $y^2=24 x^6+51 x^5+55 x^4+x^3+43 x^2+32 x+38$
• $y^2=52 x^6+x^5+2 x^4+31 x^3+59 x^2+56 x+41$
• $y^2=46 x^6+30 x^4+30 x^2+46$
• $y^2=31 x^6+5 x^5+55 x^4+34 x^3+50 x^2+20 x+49$
• $y^2=21 x^6+54 x^5+42 x^4+24 x^3+24 x^2+28 x+31$
• $y^2=3 x^6+28 x^5+54 x^4+27 x^3+17 x^2+17 x+60$
• $y^2=x^6+x^3+3$
• $y^2=15 x^6+15 x^5+23 x^4+22 x^3+23 x^2+15 x+15$
• $y^2=46 x^6+56 x^5+22 x^4+20 x^3+22 x^2+56 x+46$
• $y^2=41 x^6+33 x^5+58 x^4+13 x^3+16 x^2+44 x+27$
• $y^2=6 x^6+19 x^5+4 x^4+39 x^3+49 x^2+49 x+21$
• $y^2=x^6+4 x^3+3$
• $y^2=x^6+14 x^3+60$
• $y^2=41 x^6+54 x^5+9 x^4+42 x^3+41 x^2+43 x+1$
• $y^2=36 x^6+23 x^5+30 x^4+18 x^3+53 x^2+2 x+36$
• $y^2=10 x^6+38 x^5+8 x^4+25 x^3+23 x^2+11 x+32$
• $y^2=47 x^6+13 x^5+42 x^4+2 x^3+49 x^2+16 x+19$
• $y^2=9 x^6+8 x^5+30 x^4+7 x^3+18 x^2+59 x+20$
• $y^2=14 x^6+13 x^5+43 x^4+38 x^3+55 x^2+15 x+57$
• $y^2=31 x^6+49 x^5+12 x^4+29 x^3+12 x^2+49 x+31$
• $y^2=x^6+48 x^3+60$
• $y^2=39 x^6+49 x^5+19 x^4+43 x^3+58 x^2+46 x+22$
• $y^2=9 x^6+23 x^5+19 x^4+32 x^3+27 x^2+26 x+34$
• $y^2=16 x^6+17 x^5+36 x^4+25 x^3+12 x^2+29 x+39$
• $y^2=5 x^6+49 x^5+26 x^4+17 x^3+21 x^2+26 x+42$
• $y^2=52 x^5+35 x^4+17 x^3+28 x^2+4 x$
• $y^2=60 x^6+57 x^5+8 x^4+23 x^3+59 x^2+15 x+41$
• $y^2=15 x^6+4 x^5+59 x^4+24 x^3+24 x^2+27 x+5$
• $y^2=16 x^6+39 x^5+24 x^4+9 x^3+35 x^2+14 x+48$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{61}$.

Endomorphism algebra over $\F_{61}$

The isogeny class factors as 1.61.k^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.61.au_io	$2$	(not in LMFDB)
2.61.a_w	$2$	(not in LMFDB)
2.61.ak_bn	$3$	(not in LMFDB)