Invariants
| Level: | $240$ | $\SL_2$-level: | $240$ | Newform level: | $1$ | ||
| Index: | $360$ | $\PSL_2$-index: | $180$ | ||||
| Genus: | $14 = 1 + \frac{ 180 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $15^{2}\cdot30\cdot120$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $5 \le \gamma \le 14$ | ||||||
| $\overline{\Q}$-gonality: | $5 \le \gamma \le 14$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 120A14 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}1&210\\102&53\end{bmatrix}$, $\begin{bmatrix}15&32\\26&141\end{bmatrix}$, $\begin{bmatrix}26&113\\169&218\end{bmatrix}$, $\begin{bmatrix}50&137\\143&52\end{bmatrix}$, $\begin{bmatrix}90&167\\67&174\end{bmatrix}$, $\begin{bmatrix}124&189\\105&16\end{bmatrix}$, $\begin{bmatrix}133&20\\40&193\end{bmatrix}$, $\begin{bmatrix}224&237\\171&50\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 120.180.14.fd.1 for the level structure with $-I$) |
| Cyclic 240-isogeny field degree: | $48$ |
| Cyclic 240-torsion field degree: | $1536$ |
| Full 240-torsion field degree: | $1572864$ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $120$ | $60$ | $0$ | $0$ |
| $X_{S_4}(5)$ | $5$ | $72$ | $36$ | $0$ | $0$ |
| 16.24.0-8.n.1.8 | $16$ | $15$ | $15$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 48.72.2-24.cj.1.30 | $48$ | $5$ | $5$ | $2$ | $0$ |
| 80.120.4-40.bl.1.5 | $80$ | $3$ | $3$ | $4$ | $?$ |