Invariants
| Level: | $240$ | $\SL_2$-level: | $80$ | Newform level: | $1$ | ||
| Index: | $480$ | $\PSL_2$-index: | $240$ | ||||
| Genus: | $13 = 1 + \frac{ 240 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $5^{8}\cdot10^{4}\cdot40^{4}$ | Cusp orbits | $4^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 24$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 13$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 40G13 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}6&221\\199&132\end{bmatrix}$, $\begin{bmatrix}12&53\\7&170\end{bmatrix}$, $\begin{bmatrix}20&9\\111&14\end{bmatrix}$, $\begin{bmatrix}24&59\\91&48\end{bmatrix}$, $\begin{bmatrix}76&107\\3&44\end{bmatrix}$, $\begin{bmatrix}104&11\\69&230\end{bmatrix}$, $\begin{bmatrix}186&1\\67&64\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 120.240.13.bwr.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 no $\Q_p$ points for $p=11,17,19,83$, 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 |
|---|---|---|---|---|---|
| 15.20.0.a.1 | $15$ | $24$ | $12$ | $0$ | $0$ |
| 16.24.0-8.n.1.8 | $16$ | $20$ | $20$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 80.240.7-40.cj.1.1 | $80$ | $2$ | $2$ | $7$ | $?$ |
| 240.240.7-40.cj.1.21 | $240$ | $2$ | $2$ | $7$ | $?$ |