Invariants
Level: | $240$ | $\SL_2$-level: | $48$ | Newform level: | $1$ | ||
Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
Genus: | $7 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 20 }{2}$ | ||||||
Cusps: | $20$ (none of which are rational) | Cusp widths | $1^{4}\cdot2^{2}\cdot3^{4}\cdot4^{2}\cdot6^{2}\cdot12^{2}\cdot16^{2}\cdot48^{2}$ | Cusp orbits | $2^{6}\cdot4^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $3 \le \gamma \le 12$ | ||||||
$\overline{\Q}$-gonality: | $3 \le \gamma \le 7$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 48AP7 |
Level structure
$\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}50&31\\137&48\end{bmatrix}$, $\begin{bmatrix}57&226\\8&43\end{bmatrix}$, $\begin{bmatrix}60&23\\131&120\end{bmatrix}$, $\begin{bmatrix}70&53\\153&194\end{bmatrix}$, $\begin{bmatrix}113&48\\56&73\end{bmatrix}$, $\begin{bmatrix}204&239\\187&200\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 240.192.7.baf.2 for the level structure with $-I$) |
Cyclic 240-isogeny field degree: | $12$ |
Cyclic 240-torsion field degree: | $384$ |
Full 240-torsion field degree: | $1474560$ |
Rational points
This modular curve has no real points, and therefore no 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 |
---|---|---|---|---|---|
$X_0(3)$ | $3$ | $96$ | $48$ | $0$ | $0$ |
80.96.0-80.bo.2.2 | $80$ | $4$ | $4$ | $0$ | $?$ |
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
24.192.3-24.gf.2.3 | $24$ | $2$ | $2$ | $3$ | $0$ |
80.96.0-80.bo.2.2 | $80$ | $4$ | $4$ | $0$ | $?$ |
240.192.3-24.gf.2.13 | $240$ | $2$ | $2$ | $3$ | $?$ |
240.192.3-240.chp.2.4 | $240$ | $2$ | $2$ | $3$ | $?$ |
240.192.3-240.chp.2.25 | $240$ | $2$ | $2$ | $3$ | $?$ |
240.192.3-240.chv.1.6 | $240$ | $2$ | $2$ | $3$ | $?$ |
240.192.3-240.chv.1.69 | $240$ | $2$ | $2$ | $3$ | $?$ |