# Properties

 Label 2.61.abb_lp 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: $1 - 27 x + 301 x^{2} - 1647 x^{3} + 3721 x^{4}$ Frobenius angles: $\pm0.0643104250482$, $\pm0.230612073966$ Angle rank: $2$ (numerical) Number field: 4.0.2197.1 Galois group: $C_4$ Jacobians: 9

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 9 curves, and hence is principally polarizable:

• $y^2=43x^6+18x^5+9x^4+51x^3+58x^2+28x+55$
• $y^2=40x^6+23x^5+41x^4+46x^3+47x^2+47x+11$
• $y^2=7x^6+53x^5+56x^4+46x^3+14x^2+9x+51$
• $y^2=35x^6+59x^5+39x^4+23x^3+27x^2+51x+2$
• $y^2=16x^6+33x^5+3x^4+33x^3+25x^2+33x+5$
• $y^2=46x^6+14x^5+51x^4+6x^3+9x^2+32x+19$
• $y^2=38x^6+50x^5+5x^4+18x^3+29x^2+56x+4$
• $y^2=19x^6+32x^5+21x^4+58x^3+6x^2+11x+12$
• $y^2=37x^6+47x^5+58x^4+59x^3+48x^2+31x+45$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 2349 13382253 51465248721 191736116755173 713360334091993344 2654349124865794486533 9876826880985021723038529 36751689407267707434417008037 136753051001877533214507350737269 508858109453949447444325201111339008

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 35 3595 226739 13847923 844616930 51520377283 3142741037471 191707289788099 11694145935604151 713342911430554390

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{61}$
 The endomorphism algebra of this simple isogeny class is 4.0.2197.1.
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_lp $2$ (not in LMFDB)