Invariants
| Level: | $144$ | $\SL_2$-level: | $144$ | Newform level: | $1$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $5 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (of which $8$ are rational) | Cusp widths | $1^{6}\cdot2^{3}\cdot8^{3}\cdot9^{2}\cdot18\cdot72$ | Cusp orbits | $1^{8}\cdot2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 5$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 5$ | ||||||
| Rational cusps: | $8$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 72B5 |
Level structure
| $\GL_2(\Z/144\Z)$-generators: | $\begin{bmatrix}22&21\\15&100\end{bmatrix}$, $\begin{bmatrix}42&19\\137&68\end{bmatrix}$, $\begin{bmatrix}53&42\\130&85\end{bmatrix}$, $\begin{bmatrix}84&1\\89&68\end{bmatrix}$, $\begin{bmatrix}91&64\\92&135\end{bmatrix}$, $\begin{bmatrix}109&90\\114&13\end{bmatrix}$, $\begin{bmatrix}122&5\\79&120\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 72.144.5.bl.1 for the level structure with $-I$) |
| Cyclic 144-isogeny field degree: | $2$ |
| Cyclic 144-torsion field degree: | $48$ |
| 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 |
|---|---|---|---|---|---|
| $X_0(9)$ | $9$ | $24$ | $12$ | $0$ | $0$ |
| 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 | $48$ | $3$ | $3$ | $1$ | $0$ |