# Properties

 Label 2.61.ay_ke 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: $1 - 24 x + 264 x^{2} - 1464 x^{3} + 3721 x^{4}$ Frobenius angles: $\pm0.171237183552$, $\pm0.262983076991$ Angle rank: $2$ (numerical) Number field: 4.0.540928.1 Galois group: $D_{4}$ Jacobians: 16

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 16 curves, and hence is principally polarizable:

• $y^2=20x^6+24x^5+28x^4+12x^3+51x^2+8x+8$
• $y^2=44x^6+29x^5+45x^4+13x^3+45x^2+10x+33$
• $y^2=31x^5+56x^4+27x^3+31x^2+2x+8$
• $y^2=28x^6+21x^5+9x^4+48x^3+34x^2+46x+58$
• $y^2=59x^6+27x^5+42x^4+35x^3+44x^2+52x+22$
• $y^2=11x^6+13x^5+39x^4+45x^3+28x^2+21x+16$
• $y^2=50x^6+27x^5+53x^4+58x^3+10x^2+2x$
• $y^2=49x^6+4x^5+41x^4+9x^3+2x^2+48x+2$
• $y^2=10x^6+38x^5+30x^4+44x^3+36x^2+58x+10$
• $y^2=23x^6+50x^5+41x^4+39x^3+55x^2+8x+31$
• $y^2=8x^6+54x^5+28x^4+9x^3+35x^2+34x+44$
• $y^2=23x^6+9x^5+21x^4+5x^2+8$
• $y^2=54x^6+31x^5+58x^4+6x^3+27x^2+42x+24$
• $y^2=50x^6+47x^5+21x^4+36x^3+17x^2+14x+53$
• $y^2=8x^6+2x^5+33x^4+2x^3+12x^2+13x+51$
• $y^2=7x^6+54x^5+43x^4+54x^3+30x^2+18x+27$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 2498 13674052 51700628882 191865600966928 713414043543619298 2654366609156780277508 9876831796222033004396978 36751690937961966776395235328 136753051487209857317371980182978 508858109352033064236296786483934532

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 38 3674 227774 13857270 844680518 51520716650 3142742601470 191707297772638 11694145977106310 713342911287682874

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{61}$
 The endomorphism algebra of this simple isogeny class is 4.0.540928.1.
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_ke $2$ (not in LMFDB)