# Properties

 Label 2.61.az_km 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: $( 1 - 15 x + 61 x^{2} )( 1 - 10 x + 61 x^{2} )$ Frobenius angles: $\pm0.0900194921159$, $\pm0.278857938376$ Angle rank: $2$ (numerical) Jacobians: 15

This isogeny class is not 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 15 curves, and hence is principally polarizable:

• $y^2=25x^6+23x^5+41x^4+27x^3+36x^2+16x+53$
• $y^2=32x^6+16x^5+54x^4+28x^3+54x^2+14x+43$
• $y^2=50x^6+23x^5+15x^4+16x^3+34x^2+53x+25$
• $y^2=8x^5+37x^4+31x^3+56x^2+11x+23$
• $y^2=24x^6+38x^5+12x^4+46x^3+55x^2+12x+26$
• $y^2=26x^6+60x^5+38x^4+49x^3+4x^2+36x+36$
• $y^2=14x^6+43x^5+12x^4+18x^3+15x^2+34x+30$
• $y^2=6x^6+54x^5+2x^4+24x^3+53x^2+16x+16$
• $y^2=22x^6+14x^5+53x^4+45x^3+5x^2+12x+53$
• $y^2=30x^6+48x^5+32x^4+42x^3+9x^2+44x+31$
• $y^2=43x^6+13x^5+38x^4+19x^3+19x^2+24x+7$
• $y^2=28x^6+58x^5+38x^4+12x^3+25x^2+17x+7$
• $y^2=12x^6+5x^5+38x^4+35x^3+28x^2+59x+39$
• $y^2=54x^6+34x^5+50x^4+38x^3+45x^2+26x+8$
• $y^2=50x^6+31x^5+46x^4+18x^3+16x^2+44x+50$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 2444 13549536 51565701824 191759808240000 713351253568586204 2654339810149110859776 9876825946723658494814684 36751693269957700872277440000 136753055010413209795368992732864 508858111714715247375944979681371616

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 37 3641 227182 13849633 844606177 51520196486 3142740740197 191707309936993 11694146278385542 713342914599809681

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{61}$
 The isogeny class factors as 1.61.ap $\times$ 1.61.ak and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{61}$.

## Base change

This is a primitive isogeny class.

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.61.af_abc $2$ (not in LMFDB) 2.61.f_abc $2$ (not in LMFDB) 2.61.z_km $2$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.61.af_abc $2$ (not in LMFDB) 2.61.f_abc $2$ (not in LMFDB) 2.61.z_km $2$ (not in LMFDB) 2.61.abb_lq $4$ (not in LMFDB) 2.61.ad_acg $4$ (not in LMFDB) 2.61.d_acg $4$ (not in LMFDB) 2.61.bb_lq $4$ (not in LMFDB)