Invariants
| Level: | $6$ | $\SL_2$-level: | $6$ | ||||
| Index: | $16$ | $\PSL_2$-index: | $8$ | ||||
| Genus: | $0 = 1 + \frac{ 8 }{12} - \frac{ 0 }{4} - \frac{ 2 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $2\cdot6$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $2$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-3,-27$) |
Other labels
| Cummins and Pauli (CP) label: | 6C0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 6.16.0.4 |
Level structure
| $\GL_2(\Z/6\Z)$-generators: | $\begin{bmatrix}2&1\\3&4\end{bmatrix}$, $\begin{bmatrix}5&2\\3&1\end{bmatrix}$, $\begin{bmatrix}5&5\\0&1\end{bmatrix}$ |
| $\GL_2(\Z/6\Z)$-subgroup: | $C_3:S_3$ |
| Contains $-I$: | no $\quad$ (see 6.8.0.b.1 for the level structure with $-I$) |
| Cyclic 6-isogeny field degree: | $3$ |
| Cyclic 6-torsion field degree: | $6$ |
| Full 6-torsion field degree: | $18$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 432 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 8 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{3^3}{2^6}\cdot\frac{x^{8}(x-6y)(x-2y)^{3}(x+2y)^{3}(x+6y)}{y^{6}x^{10}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 3.8.0-3.a.1.1 | $3$ | $2$ | $2$ | $0$ | $0$ |
| 6.8.0-3.a.1.1 | $6$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 6.48.0-6.b.1.1 | $6$ | $3$ | $3$ | $0$ |
| 6.48.0-6.c.1.1 | $6$ | $3$ | $3$ | $0$ |
| 6.48.1-6.b.1.1 | $6$ | $3$ | $3$ | $1$ |
| 12.64.1-12.c.1.1 | $12$ | $4$ | $4$ | $1$ |
| 18.48.0-18.c.1.1 | $18$ | $3$ | $3$ | $0$ |
| 18.48.0-18.d.1.1 | $18$ | $3$ | $3$ | $0$ |
| 18.48.1-18.b.1.1 | $18$ | $3$ | $3$ | $1$ |
| 18.48.1-18.c.1.1 | $18$ | $3$ | $3$ | $1$ |
| 18.48.2-18.b.1.1 | $18$ | $3$ | $3$ | $2$ |
| 18.48.2-18.c.1.1 | $18$ | $3$ | $3$ | $2$ |
| 30.80.2-30.b.1.3 | $30$ | $5$ | $5$ | $2$ |
| 30.96.3-30.b.1.6 | $30$ | $6$ | $6$ | $3$ |
| 30.160.5-30.b.1.4 | $30$ | $10$ | $10$ | $5$ |
| 42.128.3-42.b.1.8 | $42$ | $8$ | $8$ | $3$ |
| 42.336.12-42.c.1.7 | $42$ | $21$ | $21$ | $12$ |
| 42.448.15-42.b.1.5 | $42$ | $28$ | $28$ | $15$ |
| 66.192.7-66.b.1.8 | $66$ | $12$ | $12$ | $7$ |
| 66.880.30-66.b.1.3 | $66$ | $55$ | $55$ | $30$ |
| 66.880.32-66.b.1.7 | $66$ | $55$ | $55$ | $32$ |
| 66.1056.39-66.b.1.8 | $66$ | $66$ | $66$ | $39$ |
| 78.224.7-78.b.1.2 | $78$ | $14$ | $14$ | $7$ |
| 102.288.11-102.b.1.6 | $102$ | $18$ | $18$ | $11$ |
| 114.320.11-114.b.1.6 | $114$ | $20$ | $20$ | $11$ |
| 138.384.15-138.b.1.8 | $138$ | $24$ | $24$ | $15$ |
| 174.480.19-174.b.1.4 | $174$ | $30$ | $30$ | $19$ |
| 186.512.19-186.b.1.8 | $186$ | $32$ | $32$ | $19$ |