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: | $10$ (none of which are rational) | Cusp widths | $2^{8}\cdot16^{2}$ | Cusp orbits | $2^{5}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16G0 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}1&224\\201&67\end{bmatrix}$, $\begin{bmatrix}39&152\\56&191\end{bmatrix}$, $\begin{bmatrix}85&24\\41&227\end{bmatrix}$, $\begin{bmatrix}129&112\\115&239\end{bmatrix}$, $\begin{bmatrix}213&8\\232&229\end{bmatrix}$, $\begin{bmatrix}223&80\\54&199\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.0.k.1 for the level structure with $-I$) |
| Cyclic 240-isogeny field degree: | $48$ |
| Cyclic 240-torsion field degree: | $3072$ |
| Full 240-torsion field degree: | $5898240$ |
Models
Smooth plane model Smooth plane model
| $ 0 $ | $=$ | $ 3 x^{2} - 2 y^{2} + 12 z^{2} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.48.0-8.q.1.4 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 240.48.0-48.f.1.2 | $240$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-48.f.1.32 | $240$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-48.f.2.3 | $240$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-48.f.2.32 | $240$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-8.q.1.3 | $240$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.