Invariants
| Level: | $240$ | $\SL_2$-level: | $16$ | Newform level: | $1152$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $5 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
| Cusps: | $24$ (none of which are rational) | Cusp widths | $4^{8}\cdot8^{12}\cdot16^{4}$ | Cusp orbits | $2^{4}\cdot4^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 5$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16O5 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}7&112\\68&159\end{bmatrix}$, $\begin{bmatrix}29&152\\180&167\end{bmatrix}$, $\begin{bmatrix}121&176\\6&95\end{bmatrix}$, $\begin{bmatrix}187&112\\110&7\end{bmatrix}$, $\begin{bmatrix}199&24\\192&119\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.192.5.m.2 for the level structure with $-I$) |
| Cyclic 240-isogeny field degree: | $48$ |
| Cyclic 240-torsion field degree: | $1536$ |
| Full 240-torsion field degree: | $1474560$ |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ - y t + z w $ |
| $=$ | $2 y^{2} + 4 y z - 2 z^{2} - w^{2} + 2 w t + t^{2}$ | |
| $=$ | $6 x^{2} + 2 y^{2} + 2 y z - w^{2} + w t$ |
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 |
|---|---|---|---|---|---|
| 40.192.1-8.j.2.3 | $40$ | $2$ | $2$ | $1$ | $0$ |
| 240.192.1-8.j.2.3 | $240$ | $2$ | $2$ | $1$ | $?$ |
| 240.192.2-48.e.2.4 | $240$ | $2$ | $2$ | $2$ | $?$ |
| 240.192.2-48.e.2.30 | $240$ | $2$ | $2$ | $2$ | $?$ |
| 240.192.2-48.f.2.4 | $240$ | $2$ | $2$ | $2$ | $?$ |
| 240.192.2-48.f.2.30 | $240$ | $2$ | $2$ | $2$ | $?$ |