Invariants
| Level: | $144$ | $\SL_2$-level: | $144$ | Newform level: | $1$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $9 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (all of which are rational) | Cusp widths | $3^{2}\cdot6\cdot9^{2}\cdot18\cdot24\cdot72$ | Cusp orbits | $1^{8}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 9$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 9$ | ||||||
| Rational cusps: | $8$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 72D9 |
Level structure
| $\GL_2(\Z/144\Z)$-generators: | $\begin{bmatrix}39&134\\68&105\end{bmatrix}$, $\begin{bmatrix}45&130\\40&111\end{bmatrix}$, $\begin{bmatrix}84&119\\103&92\end{bmatrix}$, $\begin{bmatrix}101&136\\108&1\end{bmatrix}$, $\begin{bmatrix}107&82\\80&45\end{bmatrix}$, $\begin{bmatrix}108&41\\11&138\end{bmatrix}$, $\begin{bmatrix}133&92\\24&137\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 72.144.9.dh.1 for the level structure with $-I$) |
| Cyclic 144-isogeny field degree: | $6$ |
| Cyclic 144-torsion field degree: | $144$ |
| Full 144-torsion field degree: | $331776$ |
Rational points
This modular curve has 8 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 |
|---|---|---|---|---|---|
| 9.12.0.b.1 | $9$ | $24$ | $12$ | $0$ | $0$ |
| 16.24.0-8.n.1.8 | $16$ | $12$ | $12$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 48.96.1-24.ir.1.17 | $48$ | $3$ | $3$ | $1$ | $0$ |