Invariants
| Level: | $60$ | $\SL_2$-level: | $20$ | Newform level: | $3600$ | ||
| Index: | $240$ | $\PSL_2$-index: | $120$ | ||||
| Genus: | $7 = 1 + \frac{ 120 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $10^{4}\cdot20^{4}$ | Cusp orbits | $2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20C7 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.240.7.168 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}1&0\\20&19\end{bmatrix}$, $\begin{bmatrix}1&42\\44&31\end{bmatrix}$, $\begin{bmatrix}23&14\\18&13\end{bmatrix}$, $\begin{bmatrix}47&18\\36&37\end{bmatrix}$, $\begin{bmatrix}59&40\\30&49\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.120.7.c.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $48$ |
| Cyclic 60-torsion field degree: | $768$ |
| Full 60-torsion field degree: | $9216$ |
Jacobian
| Conductor: | $2^{20}\cdot3^{8}\cdot5^{12}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{7}$ |
| Newforms: | 50.2.a.b$^{2}$, 100.2.a.a, 720.2.a.e, 720.2.a.h, 3600.2.a.l, 3600.2.a.m |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x w - x t - x u + x v + y w - y t + y u + z t - z u $ |
| $=$ | $2 x w + x t + x v + y w + 2 y v + z t + z u$ | |
| $=$ | $2 x u + x v + y t + y u + z w + z t + z u + 2 z v$ | |
| $=$ | $3 x y + 3 x z + 3 y^{2} - 3 y z - u v$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2187 x^{12} + 40824 x^{11} y + 229392 x^{10} y^{2} + 6480 x^{10} z^{2} + 358992 x^{9} y^{3} + \cdots + 27 y^{4} z^{8} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps to other modular curves
Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 10.60.3.a.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x+3y-z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -2x-y-3z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2x+y-2z$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{4}-3X^{3}Y-5X^{2}Y^{2}-4XY^{3}-2Y^{4}+3X^{3}Z-18X^{2}YZ-17XY^{2}Z+4Y^{3}Z-5X^{2}Z^{2}+17XYZ^{2}-6Y^{2}Z^{2}+4XZ^{3}+4YZ^{3}-2Z^{4} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 60.120.7.c.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 2187X^{12}+40824X^{11}Y+229392X^{10}Y^{2}+6480X^{10}Z^{2}+358992X^{9}Y^{3}+2106X^{9}YZ^{2}+134568X^{8}Y^{4}-123750X^{8}Y^{2}Z^{2}-3288X^{8}Z^{4}-52704X^{7}Y^{5}-240120X^{7}Y^{3}Z^{2}-8094X^{7}YZ^{4}-15552X^{6}Y^{6}-91368X^{6}Y^{4}Z^{2}+18645X^{6}Y^{2}Z^{4}+520X^{6}Z^{6}+1728X^{5}Y^{7}+33624X^{5}Y^{5}Z^{2}+51606X^{5}Y^{3}Z^{4}+1748X^{5}YZ^{6}+432X^{4}Y^{8}+9864X^{4}Y^{6}Z^{2}+21171X^{4}Y^{4}Z^{4}-262X^{4}Y^{2}Z^{6}-27X^{4}Z^{8}-1152X^{3}Y^{7}Z^{2}-5424X^{3}Y^{5}Z^{4}-4012X^{3}Y^{3}Z^{6}-108X^{3}YZ^{8}-288X^{2}Y^{8}Z^{2}-1584X^{2}Y^{6}Z^{4}-1986X^{2}Y^{4}Z^{6}-81X^{2}Y^{2}Z^{8}+192XY^{7}Z^{4}+24XY^{5}Z^{6}+54XY^{3}Z^{8}+48Y^{8}Z^{4}+8Y^{6}Z^{6}+27Y^{4}Z^{8} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 20.120.3-10.a.1.5 | $20$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 60.24.0-60.a.1.8 | $60$ | $10$ | $10$ | $0$ | $0$ | full Jacobian |
| 60.120.3-10.a.1.2 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.480.13-60.m.1.5 | $60$ | $2$ | $2$ | $13$ | $4$ | $1^{6}$ |
| 60.480.13-60.o.1.3 | $60$ | $2$ | $2$ | $13$ | $4$ | $1^{6}$ |
| 60.480.13-60.u.1.6 | $60$ | $2$ | $2$ | $13$ | $2$ | $1^{6}$ |
| 60.480.13-60.w.1.2 | $60$ | $2$ | $2$ | $13$ | $2$ | $1^{6}$ |
| 60.480.13-60.bk.1.2 | $60$ | $2$ | $2$ | $13$ | $4$ | $1^{6}$ |
| 60.480.13-60.bm.1.5 | $60$ | $2$ | $2$ | $13$ | $2$ | $1^{6}$ |
| 60.480.13-60.bs.1.3 | $60$ | $2$ | $2$ | $13$ | $2$ | $1^{6}$ |
| 60.480.13-60.bu.1.5 | $60$ | $2$ | $2$ | $13$ | $4$ | $1^{6}$ |
| 60.480.15-60.d.1.5 | $60$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 60.480.15-60.f.1.7 | $60$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 60.480.15-60.f.1.9 | $60$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 60.480.15-60.g.1.1 | $60$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 60.480.15-60.g.1.13 | $60$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 60.480.15-60.j.1.3 | $60$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 60.480.15-60.j.1.9 | $60$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 60.480.15-60.bc.1.2 | $60$ | $2$ | $2$ | $15$ | $3$ | $1^{8}$ |
| 60.480.15-60.bc.1.5 | $60$ | $2$ | $2$ | $15$ | $3$ | $1^{8}$ |
| 60.480.15-60.bd.1.1 | $60$ | $2$ | $2$ | $15$ | $5$ | $1^{8}$ |
| 60.480.15-60.bd.1.11 | $60$ | $2$ | $2$ | $15$ | $5$ | $1^{8}$ |
| 60.480.15-60.bg.1.5 | $60$ | $2$ | $2$ | $15$ | $3$ | $1^{8}$ |
| 60.480.15-60.bg.1.11 | $60$ | $2$ | $2$ | $15$ | $3$ | $1^{8}$ |
| 60.480.15-60.bh.1.6 | $60$ | $2$ | $2$ | $15$ | $5$ | $1^{8}$ |
| 60.480.15-60.bh.1.9 | $60$ | $2$ | $2$ | $15$ | $5$ | $1^{8}$ |
| 60.720.19-60.bo.1.3 | $60$ | $3$ | $3$ | $19$ | $5$ | $1^{12}$ |
| 60.720.27-60.cw.1.31 | $60$ | $3$ | $3$ | $27$ | $10$ | $1^{20}$ |
| 60.960.33-60.s.1.23 | $60$ | $4$ | $4$ | $33$ | $6$ | $1^{26}$ |
| 120.480.13-120.bl.1.2 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.br.1.2 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.cj.1.2 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.cp.1.2 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.ef.1.2 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.el.1.2 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.fd.1.2 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.fj.1.2 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.15-120.i.1.12 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.i.1.22 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.m.1.12 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.m.1.22 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.q.1.10 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.q.1.22 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.z.1.10 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.z.1.22 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.do.1.2 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.do.1.28 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.dr.1.2 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.dr.1.28 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.ea.1.10 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.ea.1.24 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.ed.1.10 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.ed.1.24 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |