Invariants
| Level: | $220$ | $\SL_2$-level: | $20$ | Newform level: | $1$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $5 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (of which $4$ are rational) | Cusp widths | $2^{4}\cdot4^{4}\cdot10^{4}\cdot20^{4}$ | Cusp orbits | $1^{4}\cdot2^{4}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 5$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 5$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20I5 |
Level structure
| $\GL_2(\Z/220\Z)$-generators: | $\begin{bmatrix}13&60\\214&201\end{bmatrix}$, $\begin{bmatrix}43&140\\68&151\end{bmatrix}$, $\begin{bmatrix}71&170\\146&131\end{bmatrix}$, $\begin{bmatrix}149&150\\10&61\end{bmatrix}$, $\begin{bmatrix}161&0\\174&51\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 220.144.5.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 4 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.b.1 | $44$ | $24$ | $12$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_1(2,10)$ | $10$ | $2$ | $2$ | $1$ | $0$ |
| 220.144.1-10.a.1.5 | $220$ | $2$ | $2$ | $1$ | $?$ |
| 220.144.1-220.s.2.2 | $220$ | $2$ | $2$ | $1$ | $?$ |
| 220.144.1-220.s.2.15 | $220$ | $2$ | $2$ | $1$ | $?$ |
| 220.144.3-220.cu.2.1 | $220$ | $2$ | $2$ | $3$ | $?$ |
| 220.144.3-220.cu.2.16 | $220$ | $2$ | $2$ | $3$ | $?$ |