# Properties

 Label 2.61.abb_lr 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: $1 - 27 x + 303 x^{2} - 1647 x^{3} + 3721 x^{4}$ Frobenius angles: $\pm0.114657586013$, $\pm0.208688212581$ Angle rank: $2$ (numerical) Number field: 4.0.68725.1 Galois group: $D_{4}$ Jacobians: 5

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 5 curves, and hence is principally polarizable:

• $y^2=41x^6+44x^5+17x^4+22x^3+60x^2+60x+8$
• $y^2=34x^6+32x^5+12x^4+42x^3+42x^2+36x+51$
• $y^2=42x^6+24x^5+38x^4+11x^3+55x^2+6x+54$
• $y^2=29x^6+46x^5+20x^4+12x^3+19x^2+20x+6$
• $y^2=43x^6+2x^5+54x^4+18x^3+32x^2+23x+48$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 2351 13398349 51502277891 191783445050389 713402752074698576 2654378422548308711725 9876842969484423503105891 36751696291753030273499928549 136753053010772993424220348275311 508858109538775094332170971988093184

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 35 3599 226901 13851339 844667150 51520945943 3142746156725 191707325699539 11694146107390571 713342911549467254

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{61}$
 The endomorphism algebra of this simple isogeny class is 4.0.68725.1.
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_lr $2$ (not in LMFDB)