Invariants
| Level: | $210$ | $\SL_2$-level: | $30$ | Newform level: | $1$ | ||
| Index: | $160$ | $\PSL_2$-index: | $80$ | ||||
| Genus: | $5 = 1 + \frac{ 80 }{12} - \frac{ 0 }{4} - \frac{ 2 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $10^{2}\cdot30^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $2$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 5$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 30J5 |
Level structure
| $\GL_2(\Z/210\Z)$-generators: | $\begin{bmatrix}47&181\\203&108\end{bmatrix}$, $\begin{bmatrix}85&173\\202&75\end{bmatrix}$, $\begin{bmatrix}103&204\\169&71\end{bmatrix}$, $\begin{bmatrix}150&109\\151&105\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 210.80.5.b.1 for the level structure with $-I$) |
| Cyclic 210-isogeny field degree: | $144$ |
| Cyclic 210-torsion field degree: | $6912$ |
| Full 210-torsion field degree: | $1741824$ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(5)$ | $5$ | $16$ | $8$ | $0$ | $0$ |
| 42.16.0-42.b.1.1 | $42$ | $10$ | $10$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 30.80.2-15.a.1.7 | $30$ | $2$ | $2$ | $2$ | $0$ |
| 42.16.0-42.b.1.1 | $42$ | $10$ | $10$ | $0$ | $0$ |
| 105.80.2-15.a.1.7 | $105$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 210.320.9-210.r.1.2 | $210$ | $2$ | $2$ | $9$ |
| 210.320.9-210.t.1.2 | $210$ | $2$ | $2$ | $9$ |
| 210.320.9-210.v.1.2 | $210$ | $2$ | $2$ | $9$ |
| 210.320.9-210.x.1.3 | $210$ | $2$ | $2$ | $9$ |
| 210.320.9-210.bh.1.2 | $210$ | $2$ | $2$ | $9$ |
| 210.320.9-210.bj.1.3 | $210$ | $2$ | $2$ | $9$ |
| 210.320.9-210.bl.1.2 | $210$ | $2$ | $2$ | $9$ |
| 210.320.9-210.bn.1.2 | $210$ | $2$ | $2$ | $9$ |
| 210.480.15-210.bb.1.6 | $210$ | $3$ | $3$ | $15$ |
| 210.480.15-210.bi.1.15 | $210$ | $3$ | $3$ | $15$ |
| 210.480.17-210.dr.1.1 | $210$ | $3$ | $3$ | $17$ |