# Properties

 Label 2.61.aba_kz 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: $1 - 26 x + 285 x^{2} - 1586 x^{3} + 3721 x^{4}$ Frobenius angles: $\pm0.0471438380502$, $\pm0.263959254932$ Angle rank: $2$ (numerical) Number field: 4.0.406080.2 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:

• $y^2=52x^6+24x^5+15x^4+50x^3+10x^2+12x+6$
• $y^2=60x^6+38x^5+45x^4+36x^3+19x^2+48x+40$
• $y^2=25x^6+4x^5+4x^4+50x^3+31x^2+20x+31$
• $y^2=40x^6+4x^5+22x^4+55x^3+29x^2+9$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 2395 13457505 51496571500 191723273550345 713333008278344875 2654328145571392554000 9876816969737645331500755 36751686872946600285796668105 136753051281438354671922321071500 508858110023359866417821883195332625

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 36 3616 226878 13846996 844584576 51519970078 3142737883776 191707276568356 11694145959510198 713342912228782576

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{61}$
 The endomorphism algebra of this simple isogeny class is 4.0.406080.2.
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.ba_kz $2$ (not in LMFDB)