# Properties

 Label 2.61.ay_kc 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: $( 1 - 14 x + 61 x^{2} )( 1 - 10 x + 61 x^{2} )$ Frobenius angles: $\pm0.146275019398$, $\pm0.278857938376$ Angle rank: $2$ (numerical) Jacobians: 60

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 60 curves, and hence is principally polarizable:

• $y^2=50x^6+39x^5+8x^4+25x^3+37x^2+46x+53$
• $y^2=25x^6+5x^5+56x^4+4x^3+16x^2+18x+31$
• $y^2=40x^6+52x^5+13x^3+49x^2+28x+54$
• $y^2=32x^6+45x^5+37x^4+18x^3+18x^2+52x+17$
• $y^2=55x^6+20x^5+55x^4+43x^3+55x^2+20x+55$
• $y^2=18x^6+41x^5+21x^4+28x^3+21x^2+41x+18$
• $y^2=54x^6+34x^5+17x^4+30x^3+44x^2+38x+32$
• $y^2=25x^6+39x^5+40x^4+5x^3+31x^2+41x+19$
• $y^2=10x^6+x^5+23x^4+13x^3+53x^2+9x+35$
• $y^2=10x^6+57x^5+26x^4+24x^3+49x^2+19x+39$
• $y^2=37x^6+4x^5+6x^4+5x^3+6x^2+4x+37$
• $y^2=29x^6+49x^5+39x^4+16x^3+39x^2+49x+29$
• $y^2=37x^6+59x^5+15x^4+33x^3+15x^2+59x+37$
• $y^2=54x^6+55x^5+22x^4+18x^2+4x+55$
• $y^2=21x^6+19x^5+44x^4+37x^3+49x^2+40x+54$
• $y^2=10x^6+46x^5+38x^4+59x^3+31x^2+4x+31$
• $y^2=31x^6+37x^5+x^4+52x^3+46x^2+55x+26$
• $y^2=50x^6+24x^5+36x^4+13x^3+9x^2+44x+23$
• $y^2=34x^6+24x^5+4x^4+27x^3+4x^2+24x+34$
• $y^2=15x^6+35x^5+43x^4+47x^3+39x^2+30x+33$
• and 40 more

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 2496 13658112 51667761600 191830914662400 713391542560721856 2654358551939349964800 9876832628316327858565056 36751694451781971901847961600 136753054296555381088572946814400 508858110709178338057023579214374912

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 38 3670 227630 13854766 844653878 51520560262 3142742866238 191707316101726 11694146217341510 713342913190197430

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{61}$
 The isogeny class factors as 1.61.ao $\times$ 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.ae_as $2$ (not in LMFDB) 2.61.e_as $2$ (not in LMFDB) 2.61.y_kc $2$ (not in LMFDB) 2.61.aj_ei $3$ (not in LMFDB) 2.61.d_ai $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.61.ae_as $2$ (not in LMFDB) 2.61.e_as $2$ (not in LMFDB) 2.61.y_kc $2$ (not in LMFDB) 2.61.aj_ei $3$ (not in LMFDB) 2.61.d_ai $3$ (not in LMFDB) 2.61.aba_le $4$ (not in LMFDB) 2.61.ac_abu $4$ (not in LMFDB) 2.61.c_abu $4$ (not in LMFDB) 2.61.ba_le $4$ (not in LMFDB) 2.61.ax_js $6$ (not in LMFDB) 2.61.al_fc $6$ (not in LMFDB) 2.61.ad_ai $6$ (not in LMFDB) 2.61.j_ei $6$ (not in LMFDB) 2.61.l_fc $6$ (not in LMFDB) 2.61.x_js $6$ (not in LMFDB) 2.61.az_ks $12$ (not in LMFDB) 2.61.an_fe $12$ (not in LMFDB) 2.61.al_eg $12$ (not in LMFDB) 2.61.ab_abi $12$ (not in LMFDB) 2.61.b_abi $12$ (not in LMFDB) 2.61.l_eg $12$ (not in LMFDB) 2.61.n_fe $12$ (not in LMFDB) 2.61.z_ks $12$ (not in LMFDB)