Invariants
| Level: | $210$ | $\SL_2$-level: | $14$ | ||||
| Index: | $16$ | $\PSL_2$-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 |
Level structure
| $\GL_2(\Z/210\Z)$-generators: | $\begin{bmatrix}25&28\\69&157\end{bmatrix}$, $\begin{bmatrix}32&139\\29&174\end{bmatrix}$, $\begin{bmatrix}51&200\\197&139\end{bmatrix}$, $\begin{bmatrix}152&41\\195&82\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 210.32.0-210.a.1.1, 210.32.0-210.a.1.2, 210.32.0-210.a.1.3, 210.32.0-210.a.1.4, 210.32.0-210.a.1.5, 210.32.0-210.a.1.6, 210.32.0-210.a.1.7, 210.32.0-210.a.1.8 |
| Cyclic 210-isogeny field degree: | $72$ |
| Cyclic 210-torsion field degree: | $3456$ |
| Full 210-torsion field degree: | $17418240$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_0(7)$ | $7$ | $2$ | $2$ | $0$ | $0$ |
| 30.2.0.a.1 | $30$ | $8$ | $8$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_0(7)$ | $7$ | $2$ | $2$ | $0$ | $0$ |
| 30.2.0.a.1 | $30$ | $8$ | $8$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 210.48.2.a.1 | $210$ | $3$ | $3$ | $2$ |
| 210.48.2.a.2 | $210$ | $3$ | $3$ | $2$ |
| 210.48.2.b.1 | $210$ | $3$ | $3$ | $2$ |
| 210.48.2.c.1 | $210$ | $3$ | $3$ | $2$ |
| 210.48.4.b.1 | $210$ | $3$ | $3$ | $4$ |
| 210.64.3.b.1 | $210$ | $4$ | $4$ | $3$ |
| 210.80.4.b.1 | $210$ | $5$ | $5$ | $4$ |
| 210.96.7.j.1 | $210$ | $6$ | $6$ | $7$ |
| 210.112.5.h.1 | $210$ | $7$ | $7$ | $5$ |
| 210.160.11.b.1 | $210$ | $10$ | $10$ | $11$ |