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: | $\begin{bmatrix}4&151\\89&234\end{bmatrix}$, $\begin{bmatrix}56&227\\161&178\end{bmatrix}$, $\begin{bmatrix}78&67\\157&84\end{bmatrix}$, $\begin{bmatrix}119&84\\24&203\end{bmatrix}$, $\begin{bmatrix}178&43\\167&94\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 240.48.0.cz.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.1.5 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 120.48.0-120.ej.2.8 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-16.e.1.1 | $240$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-240.p.1.1 | $240$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-240.p.1.25 | $240$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-120.ej.2.6 | $240$ | $2$ | $2$ | $0$ | $?$ |
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.by.1.15 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.cu.2.3 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.dn.2.7 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.fw.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.iw.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.jt.2.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.kb.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.la.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.lk.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.mh.2.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.mp.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.no.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.nw.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.ot.2.3 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.pb.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.qa.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.288.8-240.tv.1.35 | $240$ | $3$ | $3$ | $8$ |
| 240.384.7-240.yq.2.1 | $240$ | $4$ | $4$ | $7$ |
| 240.480.16-240.eb.2.1 | $240$ | $5$ | $5$ | $16$ |