# Properties

 Label 2.61.abb_lo 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: $1 - 27 x + 300 x^{2} - 1647 x^{3} + 3721 x^{4}$ Frobenius angles: $\pm0.0276619945946$, $\pm0.238460193285$ Angle rank: $2$ (numerical) Number field: 4.0.3757.1 Galois group: $D_{4}$ Jacobians: 4

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 4 curves, and hence is principally polarizable:

• $y^2=30x^6+50x^5+23x^4+8x^3+27x^2+54x+9$
• $y^2=31x^6+35x^5+9x^4+8x^3+57x^2+48x+7$
• $y^2=23x^6+43x^5+12x^4+60x^3+39x^2+17x+44$
• $y^2=21x^6+49x^5+53x^4+29x^3+16x^2+19x+36$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 2348 13374208 51446736848 191712370588672 713338783287918908 2654333741436015788032 9876817740850004802230012 36751684693040928601380384768 136753048779006607742829752009552 508858108414447470487634272540795648

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 35 3593 226658 13846209 844591415 51520078694 3142738129139 191707265197345 11694145745520074 713342909973328193

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{61}$
 The endomorphism algebra of this simple isogeny class is 4.0.3757.1.
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.bb_lo $2$ (not in LMFDB)