Invariants
| Level: | $36$ | $\SL_2$-level: | $18$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 6 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $6\cdot18$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $6$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-12$) |
Other labels
| Cummins and Pauli (CP) label: | 18B0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 36.48.0.3 |
Level structure
| $\GL_2(\Z/36\Z)$-generators: | $\begin{bmatrix}13&12\\27&29\end{bmatrix}$, $\begin{bmatrix}16&29\\21&22\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 36.24.0.c.1 for the level structure with $-I$) |
| Cyclic 36-isogeny field degree: | $18$ |
| Cyclic 36-torsion field degree: | $108$ |
| Full 36-torsion field degree: | $7776$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 2 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{3^3}{2^{18}}\cdot\frac{x^{24}(x^{2}+4y^{2})(x^{4}-4x^{2}y^{2}+16y^{4})(9x^{6}+64y^{6})^{3}}{y^{18}x^{30}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 9.24.0-9.b.1.2 | $9$ | $2$ | $2$ | $0$ | $0$ |
| 12.16.0-12.b.1.4 | $12$ | $3$ | $3$ | $0$ | $0$ |
| 36.24.0-9.b.1.3 | $36$ | $2$ | $2$ | $0$ | $0$ |
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.c.1.4 | $36$ | $3$ | $3$ | $2$ |
| 36.144.2-36.c.2.4 | $36$ | $3$ | $3$ | $2$ |
| 36.144.2-36.d.1.3 | $36$ | $3$ | $3$ | $2$ |
| 36.144.4-36.m.1.7 | $36$ | $3$ | $3$ | $4$ |
| 36.144.4-36.r.1.1 | $36$ | $3$ | $3$ | $4$ |
| 36.192.5-36.h.1.4 | $36$ | $4$ | $4$ | $5$ |
| 108.144.2-108.b.1.3 | $108$ | $3$ | $3$ | $2$ |
| 108.144.4-108.d.1.3 | $108$ | $3$ | $3$ | $4$ |
| 108.144.6-108.b.1.2 | $108$ | $3$ | $3$ | $6$ |
| 180.240.6-180.b.1.8 | $180$ | $5$ | $5$ | $6$ |
| 180.288.11-180.r.1.16 | $180$ | $6$ | $6$ | $11$ |
| 180.480.17-180.d.1.8 | $180$ | $10$ | $10$ | $17$ |
| 252.144.2-252.d.1.5 | $252$ | $3$ | $3$ | $2$ |
| 252.144.2-252.d.2.5 | $252$ | $3$ | $3$ | $2$ |
| 252.144.2-252.e.1.5 | $252$ | $3$ | $3$ | $2$ |
| 252.144.2-252.e.2.5 | $252$ | $3$ | $3$ | $2$ |
| 252.144.2-252.f.1.5 | $252$ | $3$ | $3$ | $2$ |
| 252.144.2-252.f.2.5 | $252$ | $3$ | $3$ | $2$ |
| 252.384.11-252.n.1.16 | $252$ | $8$ | $8$ | $11$ |