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 | $2^{8}\cdot16^{2}$ | 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: | 16G0 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}13&120\\69&91\end{bmatrix}$, $\begin{bmatrix}101&52\\190&123\end{bmatrix}$, $\begin{bmatrix}103&204\\227&103\end{bmatrix}$, $\begin{bmatrix}109&68\\78&187\end{bmatrix}$, $\begin{bmatrix}143&52\\30&17\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 240.48.0.ce.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 |
|---|---|---|---|---|---|
| 16.48.0-16.e.2.1 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 120.48.0-120.dj.1.19 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-16.e.2.9 | $240$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-240.n.2.2 | $240$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-240.n.2.33 | $240$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-120.dj.1.15 | $240$ | $2$ | $2$ | $0$ | $?$ |
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.kj.1.2 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.kk.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.kz.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.la.1.3 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.tz.1.2 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.ua.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.ux.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.uy.1.2 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bad.1.2 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bae.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bbb.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bbc.1.2 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bed.1.2 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bee.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bet.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.beu.1.2 | $240$ | $2$ | $2$ | $1$ |
| 240.288.8-240.hc.2.5 | $240$ | $3$ | $3$ | $8$ |
| 240.384.7-240.tt.1.41 | $240$ | $4$ | $4$ | $7$ |
| 240.480.16-240.dg.1.25 | $240$ | $5$ | $5$ | $16$ |