Invariants
| Level: | $252$ | $\SL_2$-level: | $36$ | 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: | $8$ (of which $4$ are rational) | Cusp widths | $6^{2}\cdot12^{2}\cdot18^{2}\cdot36^{2}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 9$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 9$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 36C9 |
Level structure
| $\GL_2(\Z/252\Z)$-generators: | $\begin{bmatrix}7&26\\60&107\end{bmatrix}$, $\begin{bmatrix}7&102\\216&67\end{bmatrix}$, $\begin{bmatrix}135&194\\76&185\end{bmatrix}$, $\begin{bmatrix}137&232\\98&69\end{bmatrix}$, $\begin{bmatrix}159&64\\202&69\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 252.144.9.f.1 for the level structure with $-I$) |
| Cyclic 252-isogeny field degree: | $48$ |
| Cyclic 252-torsion field degree: | $3456$ |
| Full 252-torsion field degree: | $2612736$ |
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 |
|---|---|---|---|---|---|
| 9.24.0-9.b.1.1 | $9$ | $12$ | $12$ | $0$ | $0$ |
| 28.12.0.b.1 | $28$ | $24$ | $12$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 18.144.4-18.b.1.1 | $18$ | $2$ | $2$ | $4$ | $0$ |
| 84.96.1-84.b.1.8 | $84$ | $3$ | $3$ | $1$ | $?$ |
| 252.144.4-18.b.1.8 | $252$ | $2$ | $2$ | $4$ | $?$ |
| 252.144.4-252.bc.1.2 | $252$ | $2$ | $2$ | $4$ | $?$ |
| 252.144.4-252.bc.1.15 | $252$ | $2$ | $2$ | $4$ | $?$ |
| 252.144.5-252.t.1.2 | $252$ | $2$ | $2$ | $5$ | $?$ |
| 252.144.5-252.t.1.15 | $252$ | $2$ | $2$ | $5$ | $?$ |