Invariants
| Level: | $240$ | $\SL_2$-level: | $48$ | Newform level: | $1$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $9 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot6^{4}\cdot12^{2}\cdot16^{2}\cdot48^{2}$ | Cusp orbits | $2^{8}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 16$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 9$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 48AO9 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}7&230\\10&51\end{bmatrix}$, $\begin{bmatrix}52&161\\145&228\end{bmatrix}$, $\begin{bmatrix}67&170\\102&71\end{bmatrix}$, $\begin{bmatrix}96&29\\29&192\end{bmatrix}$, $\begin{bmatrix}98&151\\15&34\end{bmatrix}$, $\begin{bmatrix}213&58\\152&139\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 240.192.9.ftz.1 for the level structure with $-I$) |
| Cyclic 240-isogeny field degree: | $12$ |
| Cyclic 240-torsion field degree: | $384$ |
| Full 240-torsion field degree: | $1474560$ |
Rational points
This modular curve has no real points, and therefore no 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 |
|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $96$ | $48$ | $0$ | $0$ |
| 80.96.1-80.cc.1.13 | $80$ | $4$ | $4$ | $1$ | $?$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.192.3-24.gf.2.3 | $24$ | $2$ | $2$ | $3$ | $0$ |
| 80.96.1-80.cc.1.13 | $80$ | $4$ | $4$ | $1$ | $?$ |
| 240.192.3-24.gf.2.19 | $240$ | $2$ | $2$ | $3$ | $?$ |
| 240.192.3-240.chp.2.4 | $240$ | $2$ | $2$ | $3$ | $?$ |
| 240.192.3-240.chp.2.104 | $240$ | $2$ | $2$ | $3$ | $?$ |
| 240.192.5-240.bvw.1.6 | $240$ | $2$ | $2$ | $5$ | $?$ |
| 240.192.5-240.bvw.1.60 | $240$ | $2$ | $2$ | $5$ | $?$ |