Invariants
| Level: | $120$ | $\SL_2$-level: | $12$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $1\cdot2\cdot3\cdot6$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 6F0 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}36&17\\79&20\end{bmatrix}$, $\begin{bmatrix}39&20\\70&47\end{bmatrix}$, $\begin{bmatrix}58&15\\9&118\end{bmatrix}$, $\begin{bmatrix}78&83\\23&54\end{bmatrix}$, $\begin{bmatrix}91&50\\82&81\end{bmatrix}$, $\begin{bmatrix}113&102\\108&101\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 6.12.0.a.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $24$ |
| Cyclic 120-torsion field degree: | $768$ |
| Full 120-torsion field degree: | $1474560$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 9048 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^6}\cdot\frac{x^{12}(x+2y)^{3}(x^{3}+6x^{2}y-84xy^{2}-568y^{3})^{3}}{y^{6}x^{12}(x-10y)(x+6y)^{3}(x+8y)^{2}}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 120.48.0-6.a.1.5 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-30.a.1.12 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-6.b.1.6 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-30.b.1.8 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-12.d.1.8 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-12.f.1.7 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-12.g.1.12 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-12.h.1.5 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-12.i.1.5 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-12.j.1.6 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-60.o.1.16 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-24.p.1.12 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-60.p.1.8 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-60.q.1.8 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-60.r.1.8 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-60.s.1.7 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-60.t.1.7 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-24.y.1.15 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-24.bw.1.5 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-24.bx.1.8 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-24.ca.1.16 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-24.cb.1.15 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-24.cc.1.1 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-24.cd.1.8 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.fm.1.10 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.fn.1.18 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.fo.1.29 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.fp.1.9 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.fq.1.2 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.fr.1.10 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.fs.1.32 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.ft.1.11 | $120$ | $2$ | $2$ | $0$ |
| 120.48.1-12.i.1.3 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-12.j.1.4 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-12.k.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-12.l.1.2 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-60.v.1.12 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-60.w.1.12 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-60.x.1.11 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-60.y.1.11 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-24.eq.1.7 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-24.er.1.8 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-24.es.1.7 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-24.et.1.4 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-120.iu.1.17 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-120.iv.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-120.iw.1.9 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-120.ix.1.9 | $120$ | $2$ | $2$ | $1$ |
| 120.72.0-6.a.1.3 | $120$ | $3$ | $3$ | $0$ |
| 120.120.4-30.b.1.14 | $120$ | $5$ | $5$ | $4$ |
| 120.144.3-30.a.1.58 | $120$ | $6$ | $6$ | $3$ |
| 120.240.7-30.h.1.41 | $120$ | $10$ | $10$ | $7$ |