Invariants
| Level: | $18$ | $\SL_2$-level: | $18$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 3 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $2^{3}\cdot18$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $3$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-3$) |
Other labels
| Cummins and Pauli (CP) label: | 18C0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 18.48.0.8 |
Level structure
| $\GL_2(\Z/18\Z)$-generators: | $\begin{bmatrix}10&9\\3&1\end{bmatrix}$, $\begin{bmatrix}13&2\\0&7\end{bmatrix}$, $\begin{bmatrix}17&7\\0&7\end{bmatrix}$ |
| $\GL_2(\Z/18\Z)$-subgroup: | $C_3^4:S_3$ |
| Contains $-I$: | no $\quad$ (see 18.24.0.c.1 for the level structure with $-I$) |
| Cyclic 18-isogeny field degree: | $3$ |
| Cyclic 18-torsion field degree: | $18$ |
| Full 18-torsion field degree: | $486$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 19 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2^{13}}{3^9}\cdot\frac{y^{3}x^{24}(9x^{3}+2y^{3})^{3}(9x^{3}+4y^{3})^{3}(27x^{3}+4y^{3})}{x^{42}(3x+2y)^{2}(9x^{2}-6xy+4y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 6.16.0-6.b.1.1 | $6$ | $3$ | $3$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 18.144.1-18.d.1.1 | $18$ | $3$ | $3$ | $1$ |
| 18.144.1-18.e.1.1 | $18$ | $3$ | $3$ | $1$ |
| 18.144.4-18.n.1.1 | $18$ | $3$ | $3$ | $4$ |
| 18.144.4-18.o.1.1 | $18$ | $3$ | $3$ | $4$ |
| 36.192.4-36.a.1.2 | $36$ | $4$ | $4$ | $4$ |
| 90.240.7-90.a.1.3 | $90$ | $5$ | $5$ | $7$ |
| 90.288.9-90.c.1.7 | $90$ | $6$ | $6$ | $9$ |
| 90.480.16-90.a.1.6 | $90$ | $10$ | $10$ | $16$ |
| 126.384.11-126.h.1.6 | $126$ | $8$ | $8$ | $11$ |