Invariants
| Level: | $220$ | $\SL_2$-level: | $4$ | ||||
| Index: | $12$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 4E0 |
Level structure
| $\GL_2(\Z/220\Z)$-generators: | $\begin{bmatrix}127&156\\191&169\end{bmatrix}$, $\begin{bmatrix}181&58\\137&163\end{bmatrix}$, $\begin{bmatrix}185&112\\86&151\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 220-isogeny field degree: | $144$ |
| Cyclic 220-torsion field degree: | $11520$ |
| Full 220-torsion field degree: | $50688000$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
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
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 20.6.0.b.1 | $20$ | $2$ | $2$ | $0$ | $0$ |
| 44.6.0.a.1 | $44$ | $2$ | $2$ | $0$ | $0$ |
| 220.6.0.d.1 | $220$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 220.60.4.v.1 | $220$ | $5$ | $5$ | $4$ |
| 220.72.3.bd.1 | $220$ | $6$ | $6$ | $3$ |
| 220.120.7.bl.1 | $220$ | $10$ | $10$ | $7$ |
| 220.144.9.v.1 | $220$ | $12$ | $12$ | $9$ |