Invariants
| Level: | $252$ | $\SL_2$-level: | $36$ | Newform level: | $1$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $3 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $2^{3}\cdot4^{3}\cdot18\cdot36$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 3$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 36G3 |
Level structure
| $\GL_2(\Z/252\Z)$-generators: | $\begin{bmatrix}38&29\\31&126\end{bmatrix}$, $\begin{bmatrix}54&175\\199&36\end{bmatrix}$, $\begin{bmatrix}69&88\\152&95\end{bmatrix}$, $\begin{bmatrix}119&174\\58&139\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 252.72.3.cf.1 for the level structure with $-I$) |
| Cyclic 252-isogeny field degree: | $16$ |
| Cyclic 252-torsion field degree: | $1152$ |
| 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 |
|---|---|---|---|---|---|
| 36.72.0-18.a.1.12 | $36$ | $2$ | $2$ | $0$ | $0$ |
| 84.48.1-84.p.1.7 | $84$ | $3$ | $3$ | $1$ | $?$ |
| 126.72.0-18.a.1.2 | $126$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 252.288.5-252.a.1.12 | $252$ | $2$ | $2$ | $5$ |
| 252.288.5-252.h.1.8 | $252$ | $2$ | $2$ | $5$ |
| 252.288.5-252.i.1.8 | $252$ | $2$ | $2$ | $5$ |
| 252.288.5-252.l.1.12 | $252$ | $2$ | $2$ | $5$ |
| 252.288.5-252.z.1.4 | $252$ | $2$ | $2$ | $5$ |
| 252.288.5-252.bb.1.4 | $252$ | $2$ | $2$ | $5$ |
| 252.288.5-252.bd.1.4 | $252$ | $2$ | $2$ | $5$ |
| 252.288.5-252.bf.1.4 | $252$ | $2$ | $2$ | $5$ |
| 252.432.11-252.gd.1.14 | $252$ | $3$ | $3$ | $11$ |
| 252.432.11-252.gd.2.13 | $252$ | $3$ | $3$ | $11$ |
| 252.432.11-252.gj.1.8 | $252$ | $3$ | $3$ | $11$ |
| 252.432.11-252.gj.2.8 | $252$ | $3$ | $3$ | $11$ |
| 252.432.11-252.gp.1.12 | $252$ | $3$ | $3$ | $11$ |
| 252.432.11-252.gp.2.4 | $252$ | $3$ | $3$ | $11$ |
| 252.432.11-252.gr.1.8 | $252$ | $3$ | $3$ | $11$ |
| 252.432.11-252.gr.2.8 | $252$ | $3$ | $3$ | $11$ |
| 252.432.11-252.gt.1.13 | $252$ | $3$ | $3$ | $11$ |
| 252.432.13-252.cy.1.11 | $252$ | $3$ | $3$ | $13$ |