Invariants
| Level: | $20$ | $\SL_2$-level: | $4$ | ||||
| Index: | $6$ | $\PSL_2$-index: | $6$ | ||||
| Genus: | $0 = 1 + \frac{ 6 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}$ | ||||||
| Cusps: | $3$ (of which $1$ is rational) | Cusp widths | $1^{2}\cdot4$ | Cusp orbits | $1\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $1$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 4B0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 20.6.0.4 |
Level structure
| $\GL_2(\Z/20\Z)$-generators: | $\begin{bmatrix}9&2\\6&7\end{bmatrix}$, $\begin{bmatrix}11&8\\17&7\end{bmatrix}$, $\begin{bmatrix}17&18\\17&7\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 20-isogeny field degree: | $12$ |
| Cyclic 20-torsion field degree: | $96$ |
| Full 20-torsion field degree: | $7680$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 4163 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 6 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^8}\cdot\frac{x^{6}(5x^{2}+768y^{2})^{3}}{y^{4}x^{6}(5x^{2}+1024y^{2})}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_0(2)$ | $2$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 20.12.0.a.1 | $20$ | $2$ | $2$ | $0$ |
| 20.12.0.c.1 | $20$ | $2$ | $2$ | $0$ |
| 20.12.0.j.1 | $20$ | $2$ | $2$ | $0$ |
| 20.12.0.k.1 | $20$ | $2$ | $2$ | $0$ |
| 20.30.2.f.1 | $20$ | $5$ | $5$ | $2$ |
| 20.36.1.e.1 | $20$ | $6$ | $6$ | $1$ |
| 20.60.3.n.1 | $20$ | $10$ | $10$ | $3$ |
| 40.12.0.c.1 | $40$ | $2$ | $2$ | $0$ |
| 40.12.0.h.1 | $40$ | $2$ | $2$ | $0$ |
| 40.12.0.bg.1 | $40$ | $2$ | $2$ | $0$ |
| 40.12.0.bj.1 | $40$ | $2$ | $2$ | $0$ |
| 60.12.0.i.1 | $60$ | $2$ | $2$ | $0$ |
| 60.12.0.k.1 | $60$ | $2$ | $2$ | $0$ |
| 60.12.0.bd.1 | $60$ | $2$ | $2$ | $0$ |
| 60.12.0.bf.1 | $60$ | $2$ | $2$ | $0$ |
| 60.18.1.b.1 | $60$ | $3$ | $3$ | $1$ |
| 60.24.0.p.1 | $60$ | $4$ | $4$ | $0$ |
| 120.12.0.bd.1 | $120$ | $2$ | $2$ | $0$ |
| 120.12.0.bj.1 | $120$ | $2$ | $2$ | $0$ |
| 120.12.0.dm.1 | $120$ | $2$ | $2$ | $0$ |
| 120.12.0.ds.1 | $120$ | $2$ | $2$ | $0$ |
| 140.12.0.m.1 | $140$ | $2$ | $2$ | $0$ |
| 140.12.0.n.1 | $140$ | $2$ | $2$ | $0$ |
| 140.12.0.u.1 | $140$ | $2$ | $2$ | $0$ |
| 140.12.0.v.1 | $140$ | $2$ | $2$ | $0$ |
| 140.48.2.d.1 | $140$ | $8$ | $8$ | $2$ |
| 140.126.7.b.1 | $140$ | $21$ | $21$ | $7$ |
| 140.168.9.k.1 | $140$ | $28$ | $28$ | $9$ |
| 180.162.10.b.1 | $180$ | $27$ | $27$ | $10$ |
| 220.12.0.m.1 | $220$ | $2$ | $2$ | $0$ |
| 220.12.0.n.1 | $220$ | $2$ | $2$ | $0$ |
| 220.12.0.u.1 | $220$ | $2$ | $2$ | $0$ |
| 220.12.0.v.1 | $220$ | $2$ | $2$ | $0$ |
| 220.72.4.b.1 | $220$ | $12$ | $12$ | $4$ |
| 220.330.21.b.1 | $220$ | $55$ | $55$ | $21$ |
| 220.330.21.e.1 | $220$ | $55$ | $55$ | $21$ |
| 260.12.0.m.1 | $260$ | $2$ | $2$ | $0$ |
| 260.12.0.n.1 | $260$ | $2$ | $2$ | $0$ |
| 260.12.0.u.1 | $260$ | $2$ | $2$ | $0$ |
| 260.12.0.v.1 | $260$ | $2$ | $2$ | $0$ |
| 260.84.5.b.1 | $260$ | $14$ | $14$ | $5$ |
| 280.12.0.bo.1 | $280$ | $2$ | $2$ | $0$ |
| 280.12.0.br.1 | $280$ | $2$ | $2$ | $0$ |
| 280.12.0.cm.1 | $280$ | $2$ | $2$ | $0$ |
| 280.12.0.cp.1 | $280$ | $2$ | $2$ | $0$ |