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: | $\begin{bmatrix}25&16\\34&125\end{bmatrix}$, $\begin{bmatrix}97&128\\31&37\end{bmatrix}$, $\begin{bmatrix}131&56\\60&191\end{bmatrix}$, $\begin{bmatrix}179&136\\217&211\end{bmatrix}$, $\begin{bmatrix}223&64\\140&141\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 240.48.0.ca.1 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.2.3 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 120.48.0-120.dh.1.14 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-16.e.2.16 | $240$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-240.m.2.10 | $240$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-240.m.2.57 | $240$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-120.dh.1.3 | $240$ | $2$ | $2$ | $0$ | $?$ |
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.kb.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.kc.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.kr.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.ks.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.uh.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.ui.1.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.up.1.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.uq.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bal.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bam.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bat.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bau.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bdv.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bdw.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bel.1.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bem.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.288.8-240.gu.2.5 | $240$ | $3$ | $3$ | $8$ |
| 240.384.7-240.tp.2.13 | $240$ | $4$ | $4$ | $7$ |
| 240.480.16-240.dc.1.17 | $240$ | $5$ | $5$ | $16$ |