Invariants
| Level: | $240$ | $\SL_2$-level: | $16$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $1^{4}\cdot4\cdot16$ | Cusp orbits | $1^{2}\cdot4$ | ||
| 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: | 16C0 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}37&12\\160&169\end{bmatrix}$, $\begin{bmatrix}86&171\\117&164\end{bmatrix}$, $\begin{bmatrix}131&206\\18&175\end{bmatrix}$, $\begin{bmatrix}137&196\\172&153\end{bmatrix}$, $\begin{bmatrix}158&45\\145&46\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 240.24.0.q.1 for the level structure with $-I$) |
| Cyclic 240-isogeny field degree: | $96$ |
| Cyclic 240-torsion field degree: | $3072$ |
| Full 240-torsion field degree: | $11796480$ |
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 |
|---|---|---|---|---|---|
| 8.24.0-8.o.1.3 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 240.24.0-8.o.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 |
|---|---|---|---|---|
| 240.96.1-240.b.1.2 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.c.1.4 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.n.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.o.1.4 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.z.1.4 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.ba.1.4 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.bd.1.4 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.be.1.4 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.bz.1.2 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.ca.1.2 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.cd.1.2 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.ce.1.2 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.cp.1.2 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.cq.1.2 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.ct.1.2 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.cu.1.2 | $240$ | $2$ | $2$ | $1$ |
| 240.144.4-240.co.1.47 | $240$ | $3$ | $3$ | $4$ |
| 240.192.3-240.cht.1.35 | $240$ | $4$ | $4$ | $3$ |
| 240.240.8-240.w.1.29 | $240$ | $5$ | $5$ | $8$ |
| 240.288.7-240.yk.1.43 | $240$ | $6$ | $6$ | $7$ |
| 240.480.15-240.by.1.13 | $240$ | $10$ | $10$ | $15$ |