Invariants
| Level: | $42$ | $\SL_2$-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: | none |
Other labels
| Cummins and Pauli (CP) label: | 6C0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 42.16.0.10 |
Level structure
| $\GL_2(\Z/42\Z)$-generators: | $\begin{bmatrix}8&33\\27&19\end{bmatrix}$, $\begin{bmatrix}16&35\\21&32\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 42.8.0.b.1 for the level structure with $-I$) |
| Cyclic 42-isogeny field degree: | $24$ |
| Cyclic 42-torsion field degree: | $288$ |
| Full 42-torsion field degree: | $36288$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 145 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 8 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^{12}\cdot3^3\cdot7}\cdot\frac{x^{8}(7x^{2}+144y^{2})^{3}(7x^{2}+1296y^{2})}{y^{6}x^{10}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 6.8.0-3.a.1.2 | $6$ | $2$ | $2$ | $0$ | $0$ |
| 21.8.0-3.a.1.1 | $21$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 42.48.0-42.c.1.1 | $42$ | $3$ | $3$ | $0$ |
| 42.48.1-42.c.1.2 | $42$ | $3$ | $3$ | $1$ |
| 42.128.3-42.d.1.5 | $42$ | $8$ | $8$ | $3$ |
| 42.336.12-42.e.1.6 | $42$ | $21$ | $21$ | $12$ |
| 42.448.15-42.d.1.6 | $42$ | $28$ | $28$ | $15$ |
| 84.64.1-84.c.1.3 | $84$ | $4$ | $4$ | $1$ |
| 126.48.0-126.d.1.1 | $126$ | $3$ | $3$ | $0$ |
| 126.48.1-126.b.1.1 | $126$ | $3$ | $3$ | $1$ |
| 126.48.2-126.b.1.1 | $126$ | $3$ | $3$ | $2$ |
| 210.80.2-210.b.1.7 | $210$ | $5$ | $5$ | $2$ |
| 210.96.3-210.b.1.8 | $210$ | $6$ | $6$ | $3$ |
| 210.160.5-210.b.1.3 | $210$ | $10$ | $10$ | $5$ |