Invariants
| Level: | $240$ | $\SL_2$-level: | $48$ | Newform level: | $1$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $4$ are rational) | Cusp widths | $1^{4}\cdot3^{4}\cdot4\cdot12\cdot16\cdot48$ | Cusp orbits | $1^{4}\cdot2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 3$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 48J3 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}1&128\\48&65\end{bmatrix}$, $\begin{bmatrix}10&143\\217&192\end{bmatrix}$, $\begin{bmatrix}12&1\\97&60\end{bmatrix}$, $\begin{bmatrix}93&118\\100&39\end{bmatrix}$, $\begin{bmatrix}111&50\\82&47\end{bmatrix}$, $\begin{bmatrix}217&14\\60&179\end{bmatrix}$, $\begin{bmatrix}229&138\\186&13\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 240.96.3.chw.1 for the level structure with $-I$) |
| Cyclic 240-isogeny field degree: | $12$ |
| Cyclic 240-torsion field degree: | $384$ |
| Full 240-torsion field degree: | $2949120$ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal 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 |
|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $48$ | $24$ | $0$ | $0$ |
| 80.48.0-80.p.1.3 | $80$ | $4$ | $4$ | $0$ | $?$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 48.96.1-24.ir.1.17 | $48$ | $2$ | $2$ | $1$ | $0$ |
| 80.48.0-80.p.1.3 | $80$ | $4$ | $4$ | $0$ | $?$ |
| 120.96.1-24.ir.1.40 | $120$ | $2$ | $2$ | $1$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.