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 | $1^{4}\cdot2^{2}\cdot4^{2}\cdot16^{2}$ | 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: | 16H0 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}107&144\\222&5\end{bmatrix}$, $\begin{bmatrix}190&71\\33&100\end{bmatrix}$, $\begin{bmatrix}191&122\\74&15\end{bmatrix}$, $\begin{bmatrix}197&206\\152&3\end{bmatrix}$, $\begin{bmatrix}221&88\\212&17\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 240.48.0.ex.1 for the level structure with $-I$) |
| Cyclic 240-isogeny field degree: | $24$ |
| 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.g.1.6 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 120.48.0-120.ej.1.21 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-16.g.1.7 | $240$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-240.n.2.2 | $240$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-240.n.2.9 | $240$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-120.ej.1.2 | $240$ | $2$ | $2$ | $0$ | $?$ |
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.x.1.7 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.cd.2.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.em.2.7 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.ey.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.vg.2.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.vl.1.3 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.vx.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.wa.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bbk.2.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bbp.2.3 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bcb.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bce.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bcq.2.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bdd.1.2 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.beb.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bee.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.288.8-240.zf.2.7 | $240$ | $3$ | $3$ | $8$ |
| 240.384.7-240.bdg.1.1 | $240$ | $4$ | $4$ | $7$ |
| 240.480.16-240.gr.1.2 | $240$ | $5$ | $5$ | $16$ |