# Properties

 Label 2.61.ax_ji 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: $( 1 - 15 x + 61 x^{2} )( 1 - 8 x + 61 x^{2} )$ Frobenius angles: $\pm0.0900194921159$, $\pm0.328850104905$ Angle rank: $2$ (numerical) Jacobians: 18

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 18 curves, and hence is principally polarizable:

• $y^2=31x^6+19x^5+8x^4+43x^3+48x^2+10x+31$
• $y^2=13x^6+28x^5+13x^4+42x^3+32x^2+16x+4$
• $y^2=43x^6+27x^5+15x^4+49x^3+58x^2+56x+6$
• $y^2=32x^6+2x^5+36x^4+59x^3+41x^2+45x+31$
• $y^2=35x^6+5x^5+36x^4+46x^3+18x^2+37x+8$
• $y^2=46x^6+19x^5+28x^4+53x^3+27x^2+53x+47$
• $y^2=23x^6+8x^5+13x^4+32x^3+21x^2+35x$
• $y^2=21x^6+x^5+42x^4+19x^3+28x^2+15x+23$
• $y^2=40x^6+41x^5+58x^4+52x^3+49x^2+36x+44$
• $y^2=21x^6+5x^5+57x^4+53x^3+46x^2+48x+50$
• $y^2=8x^6+51x^5+4x^4+32x^3+42x^2+32x+26$
• $y^2=40x^6+22x^5+14x^4+x^3+58x^2+49x+10$
• $y^2=7x^6+23x^5+27x^4+24x^3+10x^2+40x+59$
• $y^2=30x^6+30x^5+45x^4+56x^3+22x^2+8x+20$
• $y^2=12x^6+57x^5+27x^4+27x^3+3x^2+9x+23$
• $y^2=43x^6+54x^5+34x^4+13x^3+18x^2+2x+50$
• $y^2=16x^6+60x^5+19x^4+43x^3+19x^2+31x+2$
• $y^2=11x^6+44x^5+14x^4+23x^3+34x^2+30x+56$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 2538 13679820 51593316768 191719941336000 713313755584881858 2654328609401237556480 9876830468887493757016578 36751699363122152194506336000 136753057442086524684236978386848 508858111508917098061503113556235500

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 39 3677 227304 13846753 844561779 51519979082 3142742179119 191707341720673 11694146486324904 713342914311311477

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{61}$
 The isogeny class factors as 1.61.ap $\times$ 1.61.ai 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.ah_c $2$ (not in LMFDB) 2.61.h_c $2$ (not in LMFDB) 2.61.x_ji $2$ (not in LMFDB)