# Properties

 Label 2.61.ay_jv 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: $1 - 24 x + 255 x^{2} - 1464 x^{3} + 3721 x^{4}$ Frobenius angles: $\pm0.0628885257501$, $\pm0.312375219403$ Angle rank: $2$ (numerical) Number field: 4.0.3068560.1 Galois group: $D_{4}$ Jacobians: 8

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 8 curves, and hence is principally polarizable:

• $y^2=2x^6+53x^5+42x^4+22x^3+53x^2+52x+51$
• $y^2=26x^6+13x^5+58x^4+44x^3+56x^2+10x+38$
• $y^2=54x^6+38x^5+35x^4+37x^3+46x^2+38x+45$
• $y^2=46x^6+4x^4+34x^3+31x^2+13x+44$
• $y^2=17x^6+x^5+38x^4+4x^3+6x^2+6x+44$
• $y^2=52x^6+50x^5+12x^4+22x^3+18x^2+43x+10$
• $y^2=51x^6+24x^5+42x^4+5x^3+60x^2+41x+51$
• $y^2=14x^6+25x^5+14x^4+55x^3+30x^2+42x+13$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 2489 13602385 51552775316 191707783848265 713306406880761329 2654318595792507322000 9876821171147393177461289 36751693891826492152960349385 136753055459170272616958495253236 508858111209405594391872906389034625

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 38 3656 227126 13845876 844553078 51519784718 3142739220638 191707313180836 11694146316760046 713342913891441176

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{61}$
 The endomorphism algebra of this simple isogeny class is 4.0.3068560.1.
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_jv $2$ (not in LMFDB)