# Properties

 Label 2.61.ay_ju Base Field $\F_{61}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

# Learn more about

## Invariants

 Base field: $\F_{61}$ Dimension: $2$ L-polynomial: $1 - 24 x + 254 x^{2} - 1464 x^{3} + 3721 x^{4}$ Frobenius angles: $\pm0.0450812475667$, $\pm0.315976986299$ Angle rank: $2$ (numerical) Number field: 4.0.29952.1 Galois group: $D_{4}$ Jacobians: 14

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 14 curves, and hence is principally polarizable:

• $y^2=43x^6+31x^5+5x^4+9x^3+15x^2+6x+20$
• $y^2=32x^6+46x^5+18x^4+40x^3+26x^2+11x+10$
• $y^2=35x^6+15x^5+23x^4+35x^3+48x^2+39x+23$
• $y^2=40x^6+40x^5+39x^4+57x^3+30x^2+10x+10$
• $y^2=35x^6+42x^5+27x^4+49x^3+59x^2+9x+29$
• $y^2=28x^6+12x^5+17x^4+26x^3+50x^2+56x$
• $y^2=18x^6+x^5+40x^4+15x^3+14x^2+34x+11$
• $y^2=47x^6+58x^5+36x^4+22x^3+9x^2+36x+45$
• $y^2=50x^6+50x^5+55x^4+36x^3+57x^2+34x+12$
• $y^2=50x^6+27x^5+46x^4+32x^3+26x^2+25x+44$
• $y^2=8x^6+35x^5+45x^4+42x^3+33x^2+6x+50$
• $y^2=34x^6+48x^5+57x^4+24x^3+43x^2+44x+25$
• $y^2=7x^6+39x^5+20x^4+19x^3+42x^2+4x+26$
• $y^2=44x^6+54x^5+43x^4+23x^3+7x^2+15x+29$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 2488 13594432 51536354872 191689974125568 713293434159593848 2654311386706894028608 9876817667470287006999928 36751692069434124989762715648 136753054333130915121160458463288 508858110490563947376892091662695232

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 38 3654 227054 13844590 844537718 51519644790 3142738105790 191707303674718 11694146220469190 713342912883732774

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{61}$
 The endomorphism algebra of this simple isogeny class is 4.0.29952.1.
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_ju $2$ (not in LMFDB)