Invariants
Level: | $240$ | $\SL_2$-level: | $80$ | Newform level: | $1$ | ||
Index: | $480$ | $\PSL_2$-index: | $240$ | ||||
Genus: | $15 = 1 + \frac{ 240 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
Cusps: | $12$ (none of which are rational) | Cusp widths | $5^{4}\cdot10^{6}\cdot80^{2}$ | Cusp orbits | $2^{2}\cdot4^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $4 \le \gamma \le 28$ | ||||||
$\overline{\Q}$-gonality: | $4 \le \gamma \le 15$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 80J15 |
Level structure
$\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}17&40\\12&173\end{bmatrix}$, $\begin{bmatrix}43&128\\22&141\end{bmatrix}$, $\begin{bmatrix}58&5\\153&62\end{bmatrix}$, $\begin{bmatrix}65&214\\176&239\end{bmatrix}$, $\begin{bmatrix}152&79\\121&6\end{bmatrix}$, $\begin{bmatrix}155&102\\48&17\end{bmatrix}$, $\begin{bmatrix}235&206\\96&25\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 240.240.15.bo.2 for the level structure with $-I$) |
Cyclic 240-isogeny field degree: | $48$ |
Cyclic 240-torsion field degree: | $1536$ |
Full 240-torsion field degree: | $1179648$ |
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
The following modular covers realize this modular curve as a fiber product over $X(1)$.
Factor curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
$X_{\mathrm{ns}}^+(5)$ | $5$ | $48$ | $24$ | $0$ | $0$ |
48.48.0-48.e.1.1 | $48$ | $10$ | $10$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
48.48.0-48.e.1.1 | $48$ | $10$ | $10$ | $0$ | $0$ |
80.240.7-40.cj.1.1 | $80$ | $2$ | $2$ | $7$ | $?$ |
120.240.7-40.cj.1.7 | $120$ | $2$ | $2$ | $7$ | $?$ |