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: | $24$ (of which $3$ are rational) | Cusp widths | $3^{12}\cdot9^{4}\cdot12^{6}\cdot36^{2}$ | Cusp orbits | $1^{3}\cdot2^{4}\cdot3\cdot4\cdot6$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 7$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 7$ | ||||||
| Rational cusps: | $3$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 36N7 |
Level structure
| $\GL_2(\Z/252\Z)$-generators: | $\begin{bmatrix}37&138\\108&245\end{bmatrix}$, $\begin{bmatrix}55&237\\228&163\end{bmatrix}$, $\begin{bmatrix}143&210\\126&95\end{bmatrix}$, $\begin{bmatrix}143&222\\126&17\end{bmatrix}$, $\begin{bmatrix}251&27\\144&49\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 252.432.7-252.by.1.1, 252.432.7-252.by.1.2, 252.432.7-252.by.1.3, 252.432.7-252.by.1.4, 252.432.7-252.by.1.5, 252.432.7-252.by.1.6, 252.432.7-252.by.1.7, 252.432.7-252.by.1.8, 252.432.7-252.by.1.9, 252.432.7-252.by.1.10, 252.432.7-252.by.1.11, 252.432.7-252.by.1.12, 252.432.7-252.by.1.13, 252.432.7-252.by.1.14, 252.432.7-252.by.1.15, 252.432.7-252.by.1.16 |
| Cyclic 252-isogeny field degree: | $48$ |
| Cyclic 252-torsion field degree: | $1152$ |
| Full 252-torsion field degree: | $3483648$ |
Rational points
This modular curve has 3 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 |
|---|---|---|---|---|---|
| 18.108.2.a.1 | $18$ | $2$ | $2$ | $2$ | $0$ |
| 84.72.1.r.1 | $84$ | $3$ | $3$ | $1$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 252.432.21.d.1 | $252$ | $2$ | $2$ | $21$ |
| 252.432.21.et.1 | $252$ | $2$ | $2$ | $21$ |
| 252.432.21.rg.1 | $252$ | $2$ | $2$ | $21$ |
| 252.432.21.rj.1 | $252$ | $2$ | $2$ | $21$ |
| 252.432.21.sm.1 | $252$ | $2$ | $2$ | $21$ |
| 252.432.21.sq.1 | $252$ | $2$ | $2$ | $21$ |
| 252.432.21.sy.1 | $252$ | $2$ | $2$ | $21$ |
| 252.432.21.tb.1 | $252$ | $2$ | $2$ | $21$ |