Invariants
| Level: | $252$ | $\SL_2$-level: | $36$ | Newform level: | $1$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $4 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $3^{2}\cdot9^{2}\cdot12\cdot36$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 4$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 36F4 |
Level structure
| $\GL_2(\Z/252\Z)$-generators: | $\begin{bmatrix}166&63\\189&22\end{bmatrix}$, $\begin{bmatrix}184&111\\75&220\end{bmatrix}$, $\begin{bmatrix}205&72\\130&131\end{bmatrix}$, $\begin{bmatrix}247&62\\174&17\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 252.72.4.bc.1 for the level structure with $-I$) |
| Cyclic 252-isogeny field degree: | $48$ |
| Cyclic 252-torsion field degree: | $3456$ |
| Full 252-torsion field degree: | $5225472$ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 36.72.2-18.d.1.1 | $36$ | $2$ | $2$ | $2$ | $0$ |
| 84.48.0-84.o.1.13 | $84$ | $3$ | $3$ | $0$ | $?$ |
| 252.72.2-18.d.1.2 | $252$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 252.288.9-252.f.1.17 | $252$ | $2$ | $2$ | $9$ |
| 252.288.9-252.p.1.9 | $252$ | $2$ | $2$ | $9$ |
| 252.288.9-252.gb.1.6 | $252$ | $2$ | $2$ | $9$ |
| 252.288.9-252.gf.1.5 | $252$ | $2$ | $2$ | $9$ |
| 252.288.9-252.if.1.7 | $252$ | $2$ | $2$ | $9$ |
| 252.288.9-252.ih.1.8 | $252$ | $2$ | $2$ | $9$ |
| 252.288.9-252.ii.1.4 | $252$ | $2$ | $2$ | $9$ |
| 252.288.9-252.il.1.8 | $252$ | $2$ | $2$ | $9$ |
| 252.432.10-252.bc.1.12 | $252$ | $3$ | $3$ | $10$ |
| 252.432.10-252.dm.1.5 | $252$ | $3$ | $3$ | $10$ |
| 252.432.10-252.dm.2.13 | $252$ | $3$ | $3$ | $10$ |
| 252.432.10-252.ds.1.9 | $252$ | $3$ | $3$ | $10$ |
| 252.432.10-252.ds.2.9 | $252$ | $3$ | $3$ | $10$ |
| 252.432.10-252.dv.1.5 | $252$ | $3$ | $3$ | $10$ |
| 252.432.10-252.dv.2.5 | $252$ | $3$ | $3$ | $10$ |
| 252.432.10-252.eb.1.3 | $252$ | $3$ | $3$ | $10$ |
| 252.432.10-252.eb.2.3 | $252$ | $3$ | $3$ | $10$ |
| 252.432.10-252.ee.1.3 | $252$ | $3$ | $3$ | $10$ |