Invariants
| Level: | $252$ | $\SL_2$-level: | $36$ | Newform level: | $1$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $5 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $6\cdot12\cdot18\cdot36$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 5$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 5$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 36A5 |
Level structure
| $\GL_2(\Z/252\Z)$-generators: | $\begin{bmatrix}60&89\\127&140\end{bmatrix}$, $\begin{bmatrix}64&159\\15&136\end{bmatrix}$, $\begin{bmatrix}137&210\\240&41\end{bmatrix}$, $\begin{bmatrix}233&250\\192&121\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 252.72.5.t.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 4 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 | $36$ | $2$ | $2$ | $2$ | $0$ |
| 84.48.1-84.o.1.7 | $84$ | $3$ | $3$ | $1$ | $?$ |
| 252.72.2-18.d.1.3 | $252$ | $2$ | $2$ | $2$ | $?$ |
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.m.1.5 | $252$ | $2$ | $2$ | $9$ |
| 252.288.9-252.z.1.7 | $252$ | $2$ | $2$ | $9$ |
| 252.288.9-252.ba.1.6 | $252$ | $2$ | $2$ | $9$ |
| 252.288.9-252.gr.1.6 | $252$ | $2$ | $2$ | $9$ |
| 252.288.9-252.gw.1.5 | $252$ | $2$ | $2$ | $9$ |
| 252.288.9-252.hc.1.1 | $252$ | $2$ | $2$ | $9$ |
| 252.288.9-252.he.1.2 | $252$ | $2$ | $2$ | $9$ |
| 252.432.13-252.cx.1.11 | $252$ | $3$ | $3$ | $13$ |
| 252.432.13-252.fj.1.9 | $252$ | $3$ | $3$ | $13$ |
| 252.432.13-252.fj.2.7 | $252$ | $3$ | $3$ | $13$ |
| 252.432.13-252.fp.1.9 | $252$ | $3$ | $3$ | $13$ |
| 252.432.13-252.fp.2.9 | $252$ | $3$ | $3$ | $13$ |
| 252.432.13-252.fv.1.3 | $252$ | $3$ | $3$ | $13$ |
| 252.432.13-252.fv.2.3 | $252$ | $3$ | $3$ | $13$ |
| 252.432.13-252.fx.1.5 | $252$ | $3$ | $3$ | $13$ |
| 252.432.13-252.fx.2.5 | $252$ | $3$ | $3$ | $13$ |
| 252.432.13-252.fz.1.9 | $252$ | $3$ | $3$ | $13$ |