Properties

 Label 2.61.ay_kf 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: $( 1 - 13 x + 61 x^{2} )( 1 - 11 x + 61 x^{2} )$ Frobenius angles: $\pm0.187058313935$, $\pm0.251304563322$ Angle rank: $2$ (numerical) Jacobians: 14

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 14 curves, and hence is principally polarizable:

• $y^2=22x^6+14x^5+49x^4+16x^3+49x^2+14x+22$
• $y^2=29x^6+48x^5+2x^4+54x^3+2x^2+48x+29$
• $y^2=55x^6+45x^5+30x^4+16x^3+30x^2+45x+55$
• $y^2=28x^6+12x^5+14x^4+23x^3+14x^2+12x+28$
• $y^2=44x^6+27x^5+4x^4+27x^3+4x^2+27x+44$
• $y^2=27x^6+46x^5+12x^4+34x^3+12x^2+46x+27$
• $y^2=21x^6+41x^5+7x^4+11x^3+7x^2+41x+21$
• $y^2=48x^6+28x^5+20x^4+20x^3+20x^2+28x+48$
• $y^2=53x^6+35x^5+19x^4+11x^3+19x^2+35x+53$
• $y^2=22x^6+44x^5+5x^4+45x^3+5x^2+44x+22$
• $y^2=38x^6+27x^5+54x^4+58x^3+54x^2+27x+38$
• $y^2=11x^6+45x^5+12x^4+37x^3+12x^2+45x+11$
• $y^2=11x^6+14x^5+60x^4+7x^3+60x^2+14x+11$
• $y^2=38x^6+57x^5+4x^4+x^3+4x^2+57x+38$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 2499 13682025 51717064896 191882861829225 713424990133573179 2654370081056816640000 9876830711753278313754819 36751688609722556947197158025 136753049740527961148134288933056 508858108562305110275038089556925625

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 38 3676 227846 13858516 844693478 51520784038 3142742256398 191707285627876 11694145827742526 713342910180602476

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{61}$
 The isogeny class factors as 1.61.an $\times$ 1.61.al 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.ac_av $2$ (not in LMFDB) 2.61.c_av $2$ (not in LMFDB) 2.61.y_kf $2$ (not in LMFDB) 2.61.am_fd $3$ (not in LMFDB) 2.61.d_abg $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.61.ac_av $2$ (not in LMFDB) 2.61.c_av $2$ (not in LMFDB) 2.61.y_kf $2$ (not in LMFDB) 2.61.am_fd $3$ (not in LMFDB) 2.61.d_abg $3$ (not in LMFDB) 2.61.az_kq $6$ (not in LMFDB) 2.61.ak_eh $6$ (not in LMFDB) 2.61.ad_abg $6$ (not in LMFDB) 2.61.k_eh $6$ (not in LMFDB) 2.61.m_fd $6$ (not in LMFDB) 2.61.z_kq $6$ (not in LMFDB)