Invariants
| Level: | $240$ | $\SL_2$-level: | $240$ | Newform level: | $1$ | ||
| Index: | $432$ | $\PSL_2$-index: | $216$ | ||||
| Genus: | $15 = 1 + \frac{ 216 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (all of which are rational) | Cusp widths | $3^{2}\cdot6\cdot15^{2}\cdot24\cdot30\cdot120$ | Cusp orbits | $1^{8}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 15$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 15$ | ||||||
| Rational cusps: | $8$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 120F15 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}20&59\\71&88\end{bmatrix}$, $\begin{bmatrix}35&72\\222&125\end{bmatrix}$, $\begin{bmatrix}53&180\\6&107\end{bmatrix}$, $\begin{bmatrix}90&211\\197&24\end{bmatrix}$, $\begin{bmatrix}141&100\\112&129\end{bmatrix}$, $\begin{bmatrix}154&95\\41&128\end{bmatrix}$, $\begin{bmatrix}193&4\\100&17\end{bmatrix}$, $\begin{bmatrix}224&77\\191&190\end{bmatrix}$, $\begin{bmatrix}231&158\\158&231\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 120.216.15.hx.1 for the level structure with $-I$) |
| Cyclic 240-isogeny field degree: | $8$ |
| Cyclic 240-torsion field degree: | $256$ |
| Full 240-torsion field degree: | $1310720$ |
Rational points
This modular curve has 8 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 |
|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $144$ | $72$ | $0$ | $0$ |
| $X_0(5)$ | $5$ | $72$ | $36$ | $0$ | $0$ |
| 16.24.0-8.n.1.8 | $16$ | $18$ | $18$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 48.72.2-24.cj.1.30 | $48$ | $6$ | $6$ | $2$ | $0$ |
| 80.144.3-40.bx.1.18 | $80$ | $3$ | $3$ | $3$ | $?$ |