Invariants
| Level: | $240$ | $\SL_2$-level: | $80$ | Newform level: | $1$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $5 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $1^{4}\cdot2^{2}\cdot5^{4}\cdot8^{2}\cdot10^{2}\cdot40^{2}$ | Cusp orbits | $2^{8}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 5$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 40M5 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}7&56\\216&47\end{bmatrix}$, $\begin{bmatrix}41&178\\114&65\end{bmatrix}$, $\begin{bmatrix}44&135\\211&8\end{bmatrix}$, $\begin{bmatrix}146&221\\15&232\end{bmatrix}$, $\begin{bmatrix}179&144\\58&145\end{bmatrix}$, $\begin{bmatrix}195&98\\236&57\end{bmatrix}$, $\begin{bmatrix}226&53\\49&30\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 120.144.5.caj.2 for the level structure with $-I$) |
| Cyclic 240-isogeny field degree: | $8$ |
| Cyclic 240-torsion field degree: | $256$ |
| Full 240-torsion field degree: | $1966080$ |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=11$, 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 |
|---|---|---|---|---|---|
| 15.12.0.b.1 | $15$ | $24$ | $12$ | $0$ | $0$ |
| 16.24.0-8.n.1.8 | $16$ | $12$ | $12$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 80.144.3-40.bx.1.18 | $80$ | $2$ | $2$ | $3$ | $?$ |
| 240.144.3-40.bx.1.28 | $240$ | $2$ | $2$ | $3$ | $?$ |