Invariants
| Level: | $36$ | $\SL_2$-level: | $9$ | ||||
| Index: | $72$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $0 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 3 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $3$ are rational) | Cusp widths | $3^{3}\cdot9^{3}$ | Cusp orbits | $1^{3}\cdot3$ | ||
| Elliptic points: | $0$ of order $2$ and $3$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $3$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 9J0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 36.72.0.29 |
Level structure
| $\GL_2(\Z/36\Z)$-generators: | $\begin{bmatrix}16&1\\3&23\end{bmatrix}$, $\begin{bmatrix}26&17\\3&23\end{bmatrix}$, $\begin{bmatrix}35&31\\6&23\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 9.36.0.f.1 for the level structure with $-I$) |
| Cyclic 36-isogeny field degree: | $18$ |
| Cyclic 36-torsion field degree: | $216$ |
| 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 5 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 36 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{(x+2y)^{36}(x^{2}+xy+y^{2})^{3}(x^{3}-3x^{2}y-6xy^{2}-y^{3})(x^{9}-9x^{7}y^{2}-246x^{6}y^{3}-702x^{5}y^{4}-711x^{4}y^{5}-267x^{3}y^{6}-27x^{2}y^{7}-9xy^{8}-y^{9})^{3}}{y^{3}x^{3}(x+y)^{3}(x+2y)^{36}(x^{3}-3xy^{2}-y^{3})^{9}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 36.24.0-9.b.1.2 | $36$ | $3$ | $3$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 36.144.2-36.a.2.2 | $36$ | $2$ | $2$ | $2$ |
| 36.144.2-18.b.1.3 | $36$ | $2$ | $2$ | $2$ |
| 36.144.2-36.c.2.2 | $36$ | $2$ | $2$ | $2$ |
| 36.144.2-18.e.1.2 | $36$ | $2$ | $2$ | $2$ |
| 36.216.1-9.a.1.2 | $36$ | $3$ | $3$ | $1$ |
| 36.216.4-18.g.1.6 | $36$ | $3$ | $3$ | $4$ |
| 36.288.9-36.cr.2.7 | $36$ | $4$ | $4$ | $9$ |
| 72.144.2-72.a.2.7 | $72$ | $2$ | $2$ | $2$ |
| 72.144.2-72.b.2.2 | $72$ | $2$ | $2$ | $2$ |
| 72.144.2-72.e.2.7 | $72$ | $2$ | $2$ | $2$ |
| 72.144.2-72.f.2.3 | $72$ | $2$ | $2$ | $2$ |
| 108.216.4-27.f.1.1 | $108$ | $3$ | $3$ | $4$ |
| 108.216.7-27.c.1.1 | $108$ | $3$ | $3$ | $7$ |
| 108.216.7-27.e.1.1 | $108$ | $3$ | $3$ | $7$ |
| 180.144.2-90.a.2.2 | $180$ | $2$ | $2$ | $2$ |
| 180.144.2-180.a.2.7 | $180$ | $2$ | $2$ | $2$ |
| 180.144.2-90.b.2.2 | $180$ | $2$ | $2$ | $2$ |
| 180.144.2-180.b.1.2 | $180$ | $2$ | $2$ | $2$ |
| 180.360.11-45.a.2.4 | $180$ | $5$ | $5$ | $11$ |
| 180.432.13-45.d.2.8 | $180$ | $6$ | $6$ | $13$ |
| 252.144.2-252.g.2.2 | $252$ | $2$ | $2$ | $2$ |
| 252.144.2-252.h.2.2 | $252$ | $2$ | $2$ | $2$ |
| 252.144.2-126.j.2.1 | $252$ | $2$ | $2$ | $2$ |
| 252.144.2-126.k.1.1 | $252$ | $2$ | $2$ | $2$ |