Invariants
| Level: | $240$ | $\SL_2$-level: | $80$ | Newform level: | $1$ | ||
| Index: | $480$ | $\PSL_2$-index: | $240$ | ||||
| Genus: | $15 = 1 + \frac{ 240 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $5^{4}\cdot10^{2}\cdot20^{2}\cdot40^{4}$ | Cusp orbits | $2^{2}\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 28$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 15$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 40M15 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}29&118\\192&91\end{bmatrix}$, $\begin{bmatrix}61&190\\156&119\end{bmatrix}$, $\begin{bmatrix}136&49\\125&204\end{bmatrix}$, $\begin{bmatrix}140&147\\147&20\end{bmatrix}$, $\begin{bmatrix}165&236\\226&175\end{bmatrix}$, $\begin{bmatrix}193&204\\92&17\end{bmatrix}$, $\begin{bmatrix}216&47\\13&114\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 120.240.15.ol.1 for the level structure with $-I$) |
| Cyclic 240-isogeny field degree: | $48$ |
| Cyclic 240-torsion field degree: | $1536$ |
| Full 240-torsion field degree: | $1179648$ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known 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 |
|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(5)$ | $5$ | $48$ | $24$ | $0$ | $0$ |
| 48.48.0-24.bz.1.15 | $48$ | $10$ | $10$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 48.48.0-24.bz.1.15 | $48$ | $10$ | $10$ | $0$ | $0$ |
| 80.240.7-40.cj.1.1 | $80$ | $2$ | $2$ | $7$ | $?$ |
| 240.240.7-40.cj.1.1 | $240$ | $2$ | $2$ | $7$ | $?$ |