Invariants
| Level: | $240$ | $\SL_2$-level: | $16$ | ||||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (of which $2$ are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $1^{2}\cdot2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8O0 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}31&8\\46&31\end{bmatrix}$, $\begin{bmatrix}103&56\\174&187\end{bmatrix}$, $\begin{bmatrix}109&64\\81&113\end{bmatrix}$, $\begin{bmatrix}197&8\\141&143\end{bmatrix}$, $\begin{bmatrix}219&232\\5&107\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 120.48.0.ek.2 for the level structure with $-I$) |
| Cyclic 240-isogeny field degree: | $48$ |
| Cyclic 240-torsion field degree: | $1536$ |
| Full 240-torsion field degree: | $5898240$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 16.48.0-8.bb.2.8 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 240.48.0-8.bb.2.3 | $240$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-120.df.1.1 | $240$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-120.df.1.15 | $240$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-120.ei.2.2 | $240$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-120.ei.2.22 | $240$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 240.192.1-240.je.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.jk.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.ju.2.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.ka.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.sw.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.sy.1.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.tu.2.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.tw.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.za.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.zc.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.zy.2.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.baa.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bcy.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bde.1.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bdo.2.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bdu.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.288.8-120.tf.2.1 | $240$ | $3$ | $3$ | $8$ |
| 240.384.7-120.ms.1.27 | $240$ | $4$ | $4$ | $7$ |
| 240.480.16-120.gf.1.14 | $240$ | $5$ | $5$ | $16$ |