# Properties

 Label 2.61.ax_jk Base Field $\F_{61}$ Dimension $2$ Ordinary No $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{61}$ Dimension: $2$ L-polynomial: $1 - 23 x + 244 x^{2} - 1403 x^{3} + 3721 x^{4}$ Frobenius angles: $\pm0.109726878648$, $\pm0.321721348102$ Angle rank: $2$ (numerical) Number field: 4.0.512705.1 Galois group: $D_{4}$ Jacobians: 20

This isogeny class is simple and geometrically simple.

## Newton polygon

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

## Point counts

This isogeny class contains the Jacobians of 20 curves, and hence is principally polarizable:

• $y^2=48x^6+29x^5+60x^4+33x^3+27x^2+51x+23$
• $y^2=21x^6+18x^5+34x^3+20x^2+6x+52$
• $y^2=7x^6+11x^5+11x^4+22x^3+34x^2+20x+45$
• $y^2=16x^6+20x^5+30x^4+8x^3+54x^2+52x+33$
• $y^2=23x^6+42x^5+37x^4+31x^3+23x^2+11x+9$
• $y^2=32x^6+50x^4+26x^3+58x^2+32x+51$
• $y^2=17x^6+31x^5+54x^4+34x^3+34x^2+32x+11$
• $y^2=48x^6+27x^5+5x^4+52x^3+42x^2+55x+32$
• $y^2=46x^6+14x^5+21x^4+16x^3+50x^2+11x+54$
• $y^2=44x^6+59x^5+52x^4+13x^3+51x^2+9x+32$
• $y^2=10x^6+45x^5+38x^4+14x^3+2x^2+46x+52$
• $y^2=31x^6+30x^5+5x^4+27x^3+9x^2+49x+50$
• $y^2=42x^6+45x^5+47x^4+42x^3+49x^2+15x+59$
• $y^2=6x^6+15x^5+36x^4+55x^3+39x^2+15x+25$
• $y^2=21x^6+23x^5+27x^4+51x^3+43x^2+37x+50$
• $y^2=43x^5+34x^4+44x^3+40x^2+37x+13$
• $y^2=17x^6+41x^5+39x^4+29x^3+29x^2+16x+12$
• $y^2=46x^6+36x^5+7x^4+8x^3+13x^2+19x+41$
• $y^2=30x^6+5x^5+17x^4+58x^3+23x^2+39x+33$
• $y^2=33x^6+33x^5+9x^4+28x^3+20x^2+8x+58$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 2540 13695680 51624778640 191751626981120 713333957925433500 2654337305764156928000 9876833019031063746159260 36751699998357337855066004480 136753057761909474758084385074960 508858111703966111284402472644152000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 39 3681 227442 13849041 844585699 51520147878 3142742990559 191707345034241 11694146513673882 713342914584741001

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{61}$
 The endomorphism algebra of this simple isogeny class is 4.0.512705.1.
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.x_jk $2$ (not in LMFDB)