Invariants
| Level: | $210$ | $\SL_2$-level: | $30$ | Newform level: | $1$ | ||
| Index: | $40$ | $\PSL_2$-index: | $40$ | ||||
| Genus: | $2 = 1 + \frac{ 40 }{12} - \frac{ 0 }{4} - \frac{ 4 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $10\cdot30$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $4$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 30D2 |
Level structure
| $\GL_2(\Z/210\Z)$-generators: | $\begin{bmatrix}127&162\\193&167\end{bmatrix}$, $\begin{bmatrix}164&129\\117&143\end{bmatrix}$, $\begin{bmatrix}186&145\\55&177\end{bmatrix}$, $\begin{bmatrix}195&22\\101&175\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 210.80.2-210.c.1.1, 210.80.2-210.c.1.2, 210.80.2-210.c.1.3, 210.80.2-210.c.1.4, 210.80.2-210.c.1.5, 210.80.2-210.c.1.6, 210.80.2-210.c.1.7, 210.80.2-210.c.1.8 |
| Cyclic 210-isogeny field degree: | $144$ |
| Cyclic 210-torsion field degree: | $6912$ |
| Full 210-torsion field degree: | $6967296$ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal 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 |
|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $10$ | $10$ | $0$ | $0$ |
| 70.10.0.b.1 | $70$ | $4$ | $4$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 15.20.1.a.1 | $15$ | $2$ | $2$ | $1$ | $0$ |
| 70.10.0.b.1 | $70$ | $4$ | $4$ | $0$ | $0$ |
| 210.8.0.a.1 | $210$ | $5$ | $5$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 210.120.8.d.1 | $210$ | $3$ | $3$ | $8$ |
| 210.120.8.h.1 | $210$ | $3$ | $3$ | $8$ |
| 210.120.9.p.1 | $210$ | $3$ | $3$ | $9$ |
| 210.160.9.bc.1 | $210$ | $4$ | $4$ | $9$ |
| 210.320.23.g.1 | $210$ | $8$ | $8$ | $23$ |