# Properties

 Label 2.61.ay_jx 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: $( 1 - 15 x + 61 x^{2} )( 1 - 9 x + 61 x^{2} )$ Frobenius angles: $\pm0.0900194921159$, $\pm0.304548188780$ Angle rank: $2$ (numerical) Jacobians: 11

This isogeny class is not 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 11 curves, and hence is principally polarizable:

• $y^2=4x^6+48x^5+39x^3+48x+4$
• $y^2=8x^6+36x^5+60x^4+9x^3+48x^2+59x+59$
• $y^2=4x^6+51x^5+36x^4+44x^3+33x^2+48x+38$
• $y^2=47x^6+3x^5+60x^4+44x^3+52x^2+15x+31$
• $y^2=42x^6+54x^5+51x^4+55x^3+50x^2+11x+21$
• $y^2=53x^6+10x^5+x^4+15x^3+34x^2+7x+31$
• $y^2=40x^6+53x^5+x^4+17x^3+46x^2+27x+30$
• $y^2=x^6+33x^5+12x^4+59x^3+6x^2+2x+26$
• $y^2=11x^6+38x^5+59x^4+27x^3+59x^2+38x+11$
• $y^2=21x^6+26x^5+52x^4+5x^3+52x^2+26x+21$
• $y^2=5x^6+6x^5+17x^4+10x^3+53x^2+39x+55$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 2491 13618297 51585620800 191743238558025 713331744448229731 2654331885070267494400 9876826762457497608048331 36751696188633370639649025225 136753056669937085030627740091200 508858111958598842597433934752677737

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 38 3660 227270 13848436 844583078 51520042662 3142740999758 191707325161636 11694146420296190 713342914941697980

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{61}$
 The isogeny class factors as 1.61.ap $\times$ 1.61.aj and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
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.ag_an $2$ (not in LMFDB) 2.61.g_an $2$ (not in LMFDB) 2.61.y_jx $2$ (not in LMFDB)