Invariants
| Level: | $36$ | $\SL_2$-level: | $18$ | ||||
| Index: | $72$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $0 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $1^{3}\cdot2^{3}\cdot9\cdot18$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 18E0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 36.72.0.42 |
Level structure
| $\GL_2(\Z/36\Z)$-generators: | $\begin{bmatrix}17&22\\18&25\end{bmatrix}$, $\begin{bmatrix}19&29\\0&17\end{bmatrix}$, $\begin{bmatrix}29&6\\18&23\end{bmatrix}$, $\begin{bmatrix}29&17\\0&35\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 18.36.0.a.1 for the level structure with $-I$) |
| Cyclic 36-isogeny field degree: | $2$ |
| Cyclic 36-torsion field degree: | $24$ |
| Full 36-torsion field degree: | $5184$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 92 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 36 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{36}(x^{3}-2y^{3})^{3}(x^{9}-6x^{6}y^{3}-12x^{3}y^{6}-8y^{9})^{3}}{y^{18}x^{45}(x-2y)(x+y)^{2}(x^{2}-xy+y^{2})^{2}(x^{2}+2xy+4y^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.24.0-6.a.1.9 | $12$ | $3$ | $3$ | $0$ | $0$ |
| 36.24.0-9.a.1.1 | $36$ | $3$ | $3$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.