Invariants
| Level: | $220$ | $\SL_2$-level: | $44$ | Newform level: | $1$ | ||
| Index: | $330$ | $\PSL_2$-index: | $330$ | ||||
| Genus: | $21 = 1 + \frac{ 330 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 15 }{2}$ | ||||||
| Cusps: | $15$ (none of which are rational) | Cusp widths | $11^{10}\cdot44^{5}$ | Cusp orbits | $5\cdot10$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $6 \le \gamma \le 40$ | ||||||
| $\overline{\Q}$-gonality: | $6 \le \gamma \le 21$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 44A21 |
Level structure
| $\GL_2(\Z/220\Z)$-generators: | $\begin{bmatrix}7&106\\146&15\end{bmatrix}$, $\begin{bmatrix}44&101\\79&44\end{bmatrix}$, $\begin{bmatrix}79&192\\94&139\end{bmatrix}$, $\begin{bmatrix}162&101\\163&10\end{bmatrix}$, $\begin{bmatrix}201&80\\52&171\end{bmatrix}$ |
| 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: | $1843200$ |
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 |
|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(11)$ | $11$ | $6$ | $6$ | $1$ | $1$ |
| 20.6.0.b.1 | $20$ | $55$ | $55$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 20.6.0.b.1 | $20$ | $55$ | $55$ | $0$ | $0$ |
| 22.165.8.a.1 | $22$ | $2$ | $2$ | $8$ | $4$ |