Invariants
| Level: | $20$ | $\SL_2$-level: | $4$ | ||||
| Index: | $16$ | $\PSL_2$-index: | $8$ | ||||
| Genus: | $0 = 1 + \frac{ 8 }{12} - \frac{ 0 }{4} - \frac{ 2 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $4^{2}$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $2$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 4D0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 20.16.0.2 |
Level structure
| $\GL_2(\Z/20\Z)$-generators: | $\begin{bmatrix}10&7\\19&19\end{bmatrix}$, $\begin{bmatrix}14&13\\5&2\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 4.8.0.b.1 for the level structure with $-I$) |
| Cyclic 20-isogeny field degree: | $18$ |
| Cyclic 20-torsion field degree: | $144$ |
| Full 20-torsion field degree: | $2880$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 171 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 8 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^2}\cdot\frac{x^{8}(x^{2}-4xy-104y^{2})(x^{2}+12xy+24y^{2})^{3}}{y^{4}x^{8}(x+8y)^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 20.4.0-4.a.1.1 | $20$ | $4$ | $4$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 20.48.0-4.c.1.1 | $20$ | $3$ | $3$ | $0$ |
| 40.64.1-8.c.1.2 | $40$ | $4$ | $4$ | $1$ |
| 60.48.2-12.b.1.4 | $60$ | $3$ | $3$ | $2$ |
| 60.64.1-12.b.1.4 | $60$ | $4$ | $4$ | $1$ |
| 20.80.2-20.b.1.1 | $20$ | $5$ | $5$ | $2$ |
| 20.96.3-20.f.1.1 | $20$ | $6$ | $6$ | $3$ |
| 20.160.5-20.b.1.3 | $20$ | $10$ | $10$ | $5$ |
| 140.128.3-28.b.1.4 | $140$ | $8$ | $8$ | $3$ |
| 140.336.12-28.b.1.3 | $140$ | $21$ | $21$ | $12$ |
| 140.448.15-28.b.1.3 | $140$ | $28$ | $28$ | $15$ |
| 180.432.14-36.b.1.2 | $180$ | $27$ | $27$ | $14$ |
| 220.192.7-44.b.1.1 | $220$ | $12$ | $12$ | $7$ |
| 260.224.7-52.b.1.1 | $260$ | $14$ | $14$ | $7$ |