Invariants
| Level: | $240$ | $\SL_2$-level: | $48$ | Newform level: | $1$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $9 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $2$ are rational) | Cusp widths | $6^{4}\cdot12^{2}\cdot48^{2}$ | Cusp orbits | $1^{2}\cdot2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 9$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 9$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 48A9 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}59&218\\80&13\end{bmatrix}$, $\begin{bmatrix}83&210\\0&221\end{bmatrix}$, $\begin{bmatrix}91&128\\168&125\end{bmatrix}$, $\begin{bmatrix}151&158\\96&221\end{bmatrix}$, $\begin{bmatrix}187&230\\104&77\end{bmatrix}$, $\begin{bmatrix}195&196\\112&65\end{bmatrix}$, $\begin{bmatrix}221&140\\32&61\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 240.144.9.h.1 for the level structure with $-I$) |
| Cyclic 240-isogeny field degree: | $48$ |
| Cyclic 240-torsion field degree: | $3072$ |
| Full 240-torsion field degree: | $1966080$ |
Rational points
This modular curve has 2 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 |
|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $96$ | $48$ | $0$ | $0$ |
| 80.96.1-80.b.2.1 | $80$ | $3$ | $3$ | $1$ | $?$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.144.4-24.ch.1.38 | $24$ | $2$ | $2$ | $4$ | $0$ |
| 80.96.1-80.b.2.1 | $80$ | $3$ | $3$ | $1$ | $?$ |
| 240.144.4-24.ch.1.8 | $240$ | $2$ | $2$ | $4$ | $?$ |
| 240.144.4-240.cq.1.47 | $240$ | $2$ | $2$ | $4$ | $?$ |
| 240.144.4-240.cq.1.82 | $240$ | $2$ | $2$ | $4$ | $?$ |
| 240.144.5-240.b.1.47 | $240$ | $2$ | $2$ | $5$ | $?$ |
| 240.144.5-240.b.1.82 | $240$ | $2$ | $2$ | $5$ | $?$ |