Invariants
| Level: | $252$ | $\SL_2$-level: | $36$ | Newform level: | $1$ | ||
| Index: | $216$ | $\PSL_2$-index: | $216$ | ||||
| Genus: | $7 = 1 + \frac{ 216 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
| Cusps: | $24$ (none of which are rational) | Cusp widths | $3^{12}\cdot9^{4}\cdot12^{6}\cdot36^{2}$ | Cusp orbits | $2\cdot3^{2}\cdot4\cdot6^{2}$ | ||
| 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: | 36N7 |
Level structure
| $\GL_2(\Z/252\Z)$-generators: | $\begin{bmatrix}53&18\\108&37\end{bmatrix}$, $\begin{bmatrix}61&144\\168&137\end{bmatrix}$, $\begin{bmatrix}65&159\\138&115\end{bmatrix}$, $\begin{bmatrix}127&108\\6&199\end{bmatrix}$, $\begin{bmatrix}227&57\\48&133\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 252.432.7-252.en.1.1, 252.432.7-252.en.1.2, 252.432.7-252.en.1.3, 252.432.7-252.en.1.4, 252.432.7-252.en.1.5, 252.432.7-252.en.1.6, 252.432.7-252.en.1.7, 252.432.7-252.en.1.8 |
| Cyclic 252-isogeny field degree: | $48$ |
| Cyclic 252-torsion field degree: | $3456$ |
| Full 252-torsion field degree: | $3483648$ |
Rational points
This modular curve has no $\Q_p$ points for $p=5,41$, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 18.108.2.c.1 | $18$ | $2$ | $2$ | $2$ | $0$ |
| 84.72.1.r.1 | $84$ | $3$ | $3$ | $1$ | $?$ |
| 252.108.3.e.1 | $252$ | $2$ | $2$ | $3$ | $?$ |
| 252.108.4.f.1 | $252$ | $2$ | $2$ | $4$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 252.432.21.bp.1 | $252$ | $2$ | $2$ | $21$ |
| 252.432.21.ge.1 | $252$ | $2$ | $2$ | $21$ |
| 252.432.21.qi.1 | $252$ | $2$ | $2$ | $21$ |
| 252.432.21.qn.1 | $252$ | $2$ | $2$ | $21$ |
| 252.432.21.bem.1 | $252$ | $2$ | $2$ | $21$ |
| 252.432.21.bes.1 | $252$ | $2$ | $2$ | $21$ |
| 252.432.21.bfk.1 | $252$ | $2$ | $2$ | $21$ |
| 252.432.21.bfp.1 | $252$ | $2$ | $2$ | $21$ |