# Properties

 Label 2.61.ax_jg 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: $1 - 23 x + 240 x^{2} - 1403 x^{3} + 3721 x^{4}$ Frobenius angles: $\pm0.0670806493623$, $\pm0.335333706816$ Angle rank: $2$ (numerical) Number field: 4.0.1598508.2 Galois group: $D_{4}$ Jacobians: 12

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 12 curves, and hence is principally polarizable:

• $y^2=35x^6+17x^5+5x^4+53x^3+46x^2+7x+52$
• $y^2=43x^6+36x^5+29x^4+18x^3+50x^2+56x+32$
• $y^2=18x^6+9x^5+10x^4+53x^3+21x^2+46x+11$
• $y^2=16x^6+12x^5+30x^4+11x^3+12x^2+31x+2$
• $y^2=6x^6+52x^5+3x^4+2x^3+12x^2+34x+13$
• $y^2=59x^6+16x^5+38x^4+13x^3+11x^2+36x+36$
• $y^2=29x^6+54x^5+39x^4+21x^3+3x^2+33x+4$
• $y^2=10x^6+25x^5+20x^4+20x^3+57x^2+56x+40$
• $y^2=60x^6+55x^4+31x^3+6x^2+7x+26$
• $y^2=48x^6+41x^5+40x^4+46x^3+49x^2+52x+39$
• $y^2=30x^6+20x^5+30x^4+44x^3+45x^2+41x+11$
• $y^2=18x^6+50x^5+3x^4+4x^3+8x^2+56x+9$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 2536 13663968 51561860704 191688035816832 713292776461893256 2654318549208695334912 9876826327937573587297864 36751697348483655285774263808 136753056175367779080861619284064 508858110771447441494460621759438048

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 39 3673 227166 13844449 844536939 51519783814 3142740861495 191707331211745 11694146378004150 713342913277489393

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{61}$
 The endomorphism algebra of this simple isogeny class is 4.0.1598508.2.
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.x_jg $2$ (not in LMFDB)