# Properties

 Label 2.61.ay_jy Base Field $\F_{61}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{61}$ Dimension: $2$ L-polynomial: $1 - 24 x + 258 x^{2} - 1464 x^{3} + 3721 x^{4}$ Frobenius angles: $\pm0.101801111500$, $\pm0.300249954030$ Angle rank: $2$ (numerical) Number field: 4.0.15424.2 Galois group: $D_{4}$ Jacobians: 40

This isogeny class is simple and geometrically simple.

## Newton polygon

This isogeny class is ordinary.

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

## Point counts

This isogeny class contains the Jacobians of 40 curves, and hence is principally polarizable:

• $y^2=35x^6+60x^5+41x^4+24x^3+60x^2+5x+9$
• $y^2=40x^6+50x^5+60x^4+19x^3+32x^2+11x+17$
• $y^2=54x^6+35x^5+5x^4+x^3+50x^2+26x+38$
• $y^2=41x^6+21x^5+21x^4+36x^3+11x^2+19x+14$
• $y^2=31x^6+53x^5+22x^4+11x^3+47x^2+43$
• $y^2=52x^6+25x^5+14x^4+53x^3+43x^2+44x+34$
• $y^2=38x^6+12x^5+11x^4+34x^3+43x^2+38x+44$
• $y^2=11x^5+6x^4+9x^3+8x^2+30x+22$
• $y^2=18x^6+54x^5+52x^4+24x^3+57x^2+39x+29$
• $y^2=32x^6+25x^5+8x^4+3x^3+6x^2+36x+18$
• $y^2=19x^6+26x^5+21x^4+45x^3+23x^2+57x+56$
• $y^2=x^6+4x^5+24x^4+52x^3+53x^2+49x+38$
• $y^2=2x^6+51x^5+20x^4+53x^3+13x^2+32x+55$
• $y^2=35x^6+59x^5+52x^4+38x^3+9x^2+13x+44$
• $y^2=42x^6+24x^5+34x^4+39x^3+3x^2+x+23$
• $y^2=12x^6+33x^5+54x^4+2x^3+36x^2+48x+35$
• $y^2=47x^6+26x^5+35x^4+32x^3+46x^2+16x+59$
• $y^2=47x^6+27x^4+46x^3+57x^2+39x+55$
• $y^2=48x^6+14x^5+8x^4+14x^3+24x^2+59x+31$
• $y^2=22x^6+53x^5+60x^4+60x^3+34x^2+35x+58$
• and 20 more

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 2492 13626256 51602045852 191760883557376 713344109300063612 2654337966500373767824 9876828856426519717253468 36751696679537806726511345664 136753056783672385541733944655932 508858112027660653105702297431768976

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 38 3662 227342 13849710 844597718 51520160702 3142741666046 191707327722334 11694146430022022 713342915038512302

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{61}$
 The endomorphism algebra of this simple isogeny class is 4.0.15424.2.
All geometric endomorphisms are defined over $\F_{61}$.

## 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_jy $2$ (not in LMFDB)