# Properties

 Label 2.61.ax_jf 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: $1 - 23 x + 239 x^{2} - 1403 x^{3} + 3721 x^{4}$ Frobenius angles: $\pm0.0529194219962$, $\pm0.338378256587$ Angle rank: $2$ (numerical) Number field: 4.0.4632645.1 Galois group: $D_{4}$ Jacobians: 8

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 8 curves, and hence is principally polarizable:

• $y^2=42x^6+9x^5+54x^4+9x^3+54x^2+24x+35$
• $y^2=51x^6+16x^5+39x^4+53x^2+40x+35$
• $y^2=47x^6+12x^5+16x^4+26x^3+32x^2+14x+28$
• $y^2=24x^6+45x^5+49x^4+60x^3+29x^2+2x+55$
• $y^2=53x^6+57x^5+31x^4+2x^3+14x^2+13x+59$
• $y^2=21x^6+14x^5+27x^4+51x^3+8x^2+38x+11$
• $y^2=28x^6+13x^5+25x^4+51x^3+8x^2+16x+8$
• $y^2=24x^6+17x^5+9x^4+58x^3+49x^2+50x+37$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 2535 13656045 51546134835 191672000590005 713281995600954000 2654313006129263344125 9876823653320296055856915 36751695805098913945984889445 136753055155719811444590130269015 508858110160275873376900208127648000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 39 3671 227097 13843291 844524174 51519676223 3142740010449 191707323161011 11694146290811127 713342912420718326

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{61}$
 The endomorphism algebra of this simple isogeny class is 4.0.4632645.1.
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.x_jf $2$ (not in LMFDB)