Invariants
| Level: | $42$ | $\SL_2$-level: | $14$ | ||||
| Index: | $32$ | $\PSL_2$-index: | $16$ | ||||
| Genus: | $0 = 1 + \frac{ 16 }{12} - \frac{ 0 }{4} - \frac{ 4 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $2\cdot14$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $4$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 14B0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 42.32.0.6 |
Level structure
| $\GL_2(\Z/42\Z)$-generators: | $\begin{bmatrix}7&4\\30&5\end{bmatrix}$, $\begin{bmatrix}23&18\\6&41\end{bmatrix}$, $\begin{bmatrix}29&31\\23&30\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 14.16.0.a.1 for the level structure with $-I$) |
| Cyclic 42-isogeny field degree: | $12$ |
| Cyclic 42-torsion field degree: | $144$ |
| Full 42-torsion field degree: | $18144$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 12 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 16 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^2\cdot3^2}\cdot\frac{x^{16}(7x^{2}-6xy+36y^{2})(7x^{2}+6xy+36y^{2})(x^{4}+180x^{2}y^{2}+1296y^{4})^{3}}{y^{2}x^{30}}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_{\mathrm{ns}}(2)$ | $2$ | $16$ | $8$ | $0$ | $0$ |
| 21.16.0-7.a.1.1 | $21$ | $2$ | $2$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 21.16.0-7.a.1.1 | $21$ | $2$ | $2$ | $0$ | $0$ |
| 42.16.0-7.a.1.1 | $42$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 42.96.0-14.a.1.1 | $42$ | $3$ | $3$ | $0$ |
| 42.96.0-14.b.1.1 | $42$ | $3$ | $3$ | $0$ |
| 42.96.0-14.b.2.1 | $42$ | $3$ | $3$ | $0$ |
| 42.96.2-14.a.1.2 | $42$ | $3$ | $3$ | $2$ |
| 42.96.2-14.b.1.1 | $42$ | $3$ | $3$ | $2$ |
| 42.96.2-14.c.1.2 | $42$ | $3$ | $3$ | $2$ |
| 42.96.2-14.c.2.1 | $42$ | $3$ | $3$ | $2$ |
| 42.96.2-14.d.1.1 | $42$ | $3$ | $3$ | $2$ |
| 42.96.4-42.a.1.6 | $42$ | $3$ | $3$ | $4$ |
| 42.128.3-42.a.1.6 | $42$ | $4$ | $4$ | $3$ |
| 42.224.5-14.a.1.1 | $42$ | $7$ | $7$ | $5$ |
| 84.64.0-28.a.1.2 | $84$ | $2$ | $2$ | $0$ |
| 84.64.0-28.a.2.1 | $84$ | $2$ | $2$ | $0$ |
| 84.64.0-84.a.1.4 | $84$ | $2$ | $2$ | $0$ |
| 84.64.0-84.a.2.8 | $84$ | $2$ | $2$ | $0$ |
| 84.128.3-28.a.1.5 | $84$ | $4$ | $4$ | $3$ |
| 126.96.0-126.a.1.1 | $126$ | $3$ | $3$ | $0$ |
| 126.96.0-126.a.2.3 | $126$ | $3$ | $3$ | $0$ |
| 126.96.0-126.b.1.1 | $126$ | $3$ | $3$ | $0$ |
| 126.96.0-126.b.2.1 | $126$ | $3$ | $3$ | $0$ |
| 126.96.0-126.c.1.1 | $126$ | $3$ | $3$ | $0$ |
| 126.96.0-126.c.2.3 | $126$ | $3$ | $3$ | $0$ |
| 126.96.2-126.a.1.1 | $126$ | $3$ | $3$ | $2$ |
| 126.96.2-126.b.1.1 | $126$ | $3$ | $3$ | $2$ |
| 126.96.2-126.c.1.2 | $126$ | $3$ | $3$ | $2$ |
| 126.96.2-126.d.1.2 | $126$ | $3$ | $3$ | $2$ |
| 126.96.2-126.d.2.2 | $126$ | $3$ | $3$ | $2$ |
| 126.96.2-126.e.1.3 | $126$ | $3$ | $3$ | $2$ |
| 126.96.2-126.e.2.1 | $126$ | $3$ | $3$ | $2$ |
| 126.96.2-126.f.1.3 | $126$ | $3$ | $3$ | $2$ |
| 126.96.2-126.f.2.1 | $126$ | $3$ | $3$ | $2$ |
| 168.64.0-56.a.1.7 | $168$ | $2$ | $2$ | $0$ |
| 168.64.0-56.a.2.8 | $168$ | $2$ | $2$ | $0$ |
| 168.64.0-168.a.1.4 | $168$ | $2$ | $2$ | $0$ |
| 168.64.0-168.a.2.8 | $168$ | $2$ | $2$ | $0$ |
| 210.160.4-70.a.1.3 | $210$ | $5$ | $5$ | $4$ |
| 210.192.7-70.a.1.1 | $210$ | $6$ | $6$ | $7$ |
| 210.320.11-70.a.1.4 | $210$ | $10$ | $10$ | $11$ |
| 294.224.5-98.a.1.2 | $294$ | $7$ | $7$ | $5$ |