Invariants
| Level: | $6$ | $\SL_2$-level: | $6$ | ||||
| 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 | $3^{4}\cdot6^{4}$ | 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: | 6K0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 6.72.0.1 |
Level structure
| $\GL_2(\Z/6\Z)$-generators: | $\begin{bmatrix}1&3\\0&1\end{bmatrix}$, $\begin{bmatrix}5&0\\0&1\end{bmatrix}$ |
| $\GL_2(\Z/6\Z)$-subgroup: | $C_2^2$ |
| Contains $-I$: | no $\quad$ (see 6.36.0.a.1 for the level structure with $-I$) |
| Cyclic 6-isogeny field degree: | $1$ |
| Cyclic 6-torsion field degree: | $1$ |
| Full 6-torsion field degree: | $4$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 46 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 36 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{36}(x^{3}-2y^{3})^{3}(x^{3}+6xy^{2}-2y^{3})^{3}(x^{6}-6x^{4}y^{2}-4x^{3}y^{3}+36x^{2}y^{4}+12xy^{5}+4y^{6})^{3}}{y^{6}x^{39}(x-2y)^{3}(x+y)^{6}(x^{2}-xy+y^{2})^{6}(x^{2}+2xy+4y^{2})^{3}}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_0(2)$ | $2$ | $24$ | $12$ | $0$ | $0$ |
| $X_{\mathrm{arith}}(3)$ | $3$ | $3$ | $3$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_{\mathrm{arith}}(3)$ | $3$ | $3$ | $3$ | $0$ | $0$ |
| 6.24.0-6.a.1.2 | $6$ | $3$ | $3$ | $0$ | $0$ |
| $X_1(6)$ | $6$ | $3$ | $3$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.