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 | $5^{4}\cdot10^{6}\cdot80^{2}$ | 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: | 80J15 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}8&51\\69&14\end{bmatrix}$, $\begin{bmatrix}58&103\\41&72\end{bmatrix}$, $\begin{bmatrix}87&98\\112&105\end{bmatrix}$, $\begin{bmatrix}104&191\\75&116\end{bmatrix}$, $\begin{bmatrix}149&230\\220&119\end{bmatrix}$, $\begin{bmatrix}173&112\\128&125\end{bmatrix}$, $\begin{bmatrix}214&147\\63&106\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 240.240.15.bo.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 |
|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(5)$ | $5$ | $48$ | $24$ | $0$ | $0$ |
| 48.48.0-48.e.2.1 | $48$ | $10$ | $10$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 48.48.0-48.e.2.1 | $48$ | $10$ | $10$ | $0$ | $0$ |
| 80.240.7-40.cj.1.1 | $80$ | $2$ | $2$ | $7$ | $?$ |
| 120.240.7-40.cj.1.28 | $120$ | $2$ | $2$ | $7$ | $?$ |