Invariants
| Level: | $210$ | $\SL_2$-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 |
Level structure
| $\GL_2(\Z/210\Z)$-generators: | $\begin{bmatrix}137&161\\196&39\end{bmatrix}$, $\begin{bmatrix}159&92\\64&131\end{bmatrix}$, $\begin{bmatrix}169&131\\141&200\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 70.16.0.a.1 for the level structure with $-I$) |
| Cyclic 210-isogeny field degree: | $72$ |
| Cyclic 210-torsion field degree: | $3456$ |
| Full 210-torsion field degree: | $8709120$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 2 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 16 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^{42}\cdot3^{14}\cdot5}\cdot\frac{x^{16}(25x^{4}+14400x^{2}y^{2}+331776y^{4})^{3}(25x^{4}+37440x^{2}y^{2}+16257024y^{4})}{y^{14}x^{18}}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 10.2.0.a.1 | $10$ | $16$ | $8$ | $0$ | $0$ |
| 21.16.0-7.a.1.2 | $21$ | $2$ | $2$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 21.16.0-7.a.1.2 | $21$ | $2$ | $2$ | $0$ | $0$ |
| 210.16.0-7.a.1.2 | $210$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 210.96.2-70.a.1.3 | $210$ | $3$ | $3$ | $2$ |
| 210.96.2-70.a.2.1 | $210$ | $3$ | $3$ | $2$ |
| 210.96.2-70.b.1.2 | $210$ | $3$ | $3$ | $2$ |
| 210.96.2-70.c.1.1 | $210$ | $3$ | $3$ | $2$ |
| 210.96.4-210.a.1.12 | $210$ | $3$ | $3$ | $4$ |
| 210.128.3-210.a.1.7 | $210$ | $4$ | $4$ | $3$ |
| 210.160.4-70.b.1.3 | $210$ | $5$ | $5$ | $4$ |
| 210.192.7-70.b.1.2 | $210$ | $6$ | $6$ | $7$ |
| 210.224.5-70.c.1.2 | $210$ | $7$ | $7$ | $5$ |
| 210.320.11-70.b.1.6 | $210$ | $10$ | $10$ | $11$ |