Invariants
| Level: | $220$ | $\SL_2$-level: | $20$ | Newform level: | $1$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $7 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $6$ are rational) | Cusp widths | $4^{6}\cdot20^{6}$ | Cusp orbits | $1^{6}\cdot2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 7$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 7$ | ||||||
| Rational cusps: | $6$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20J7 |
Level structure
| $\GL_2(\Z/220\Z)$-generators: | $\begin{bmatrix}13&62\\166&17\end{bmatrix}$, $\begin{bmatrix}35&104\\218&189\end{bmatrix}$, $\begin{bmatrix}37&196\\214&75\end{bmatrix}$, $\begin{bmatrix}69&118\\92&193\end{bmatrix}$, $\begin{bmatrix}131&30\\0&161\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 220.144.7.b.1 for the level structure with $-I$) |
| Cyclic 220-isogeny field degree: | $24$ |
| Cyclic 220-torsion field degree: | $480$ |
| Full 220-torsion field degree: | $2112000$ |
Rational points
This modular curve has 6 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 |
|---|---|---|---|---|---|
| $X_1(5)$ | $5$ | $12$ | $12$ | $0$ | $0$ |
| 44.12.0-2.a.1.2 | $44$ | $24$ | $24$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_1(2,10)$ | $10$ | $2$ | $2$ | $1$ | $0$ |
| 220.96.3-220.c.1.7 | $220$ | $3$ | $3$ | $3$ | $?$ |
| 220.144.1-10.a.1.9 | $220$ | $2$ | $2$ | $1$ | $?$ |