Invariants
| Level: | $240$ | $\SL_2$-level: | $48$ | Newform level: | $1$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $5 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $2^{2}\cdot4\cdot6^{2}\cdot12\cdot16\cdot48$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 5$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 5$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 48D5 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}99&68\\88&215\end{bmatrix}$, $\begin{bmatrix}119&28\\140&207\end{bmatrix}$, $\begin{bmatrix}134&81\\209&130\end{bmatrix}$, $\begin{bmatrix}200&163\\179&12\end{bmatrix}$, $\begin{bmatrix}207&116\\74&225\end{bmatrix}$, $\begin{bmatrix}218&21\\9&194\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 240.96.5.bvy.1 for the level structure with $-I$) |
| Cyclic 240-isogeny field degree: | $24$ |
| Cyclic 240-torsion field degree: | $1536$ |
| Full 240-torsion field degree: | $2949120$ |
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 |
|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $48$ | $24$ | $0$ | $0$ |
| 80.48.1-80.d.1.14 | $80$ | $4$ | $4$ | $1$ | $?$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 48.96.1-24.iw.1.7 | $48$ | $2$ | $2$ | $1$ | $0$ |
| 80.48.1-80.d.1.14 | $80$ | $4$ | $4$ | $1$ | $?$ |
| 120.96.1-24.iw.1.24 | $120$ | $2$ | $2$ | $1$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.