Invariants
| Level: | $20$ | $\SL_2$-level: | $5$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $1^{2}\cdot5^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 5D0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 20.24.0.37 |
Level structure
| $\GL_2(\Z/20\Z)$-generators: | $\begin{bmatrix}10&19\\7&17\end{bmatrix}$, $\begin{bmatrix}13&4\\9&9\end{bmatrix}$, $\begin{bmatrix}13&15\\9&6\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 5.12.0.a.2 for the level structure with $-I$) |
| Cyclic 20-isogeny field degree: | $6$ |
| Cyclic 20-torsion field degree: | $48$ |
| Full 20-torsion field degree: | $1920$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 1545 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{3^2\cdot7}\cdot\frac{(105x+11y)^{12}(388962000x^{4}+157437000x^{3}y+23549400x^{2}y^{2}+1543920xy^{3}+37469y^{4})^{3}}{(15x+2y)(35x+3y)(105x+11y)^{12}(2205x^{2}+525xy+29y^{2})^{5}}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 20.120.0-5.a.1.1 | $20$ | $5$ | $5$ | $0$ |
| 20.48.1-10.a.2.1 | $20$ | $2$ | $2$ | $1$ |
| 20.48.1-10.b.2.1 | $20$ | $2$ | $2$ | $1$ |
| 20.72.0-10.a.1.7 | $20$ | $3$ | $3$ | $0$ |
| 60.72.2-15.a.2.8 | $60$ | $3$ | $3$ | $2$ |
| 60.96.1-15.a.2.4 | $60$ | $4$ | $4$ | $1$ |
| 20.48.1-20.b.2.2 | $20$ | $2$ | $2$ | $1$ |
| 20.48.1-20.e.2.2 | $20$ | $2$ | $2$ | $1$ |
| 20.96.3-20.i.2.6 | $20$ | $4$ | $4$ | $3$ |
| 100.120.0-25.a.2.1 | $100$ | $5$ | $5$ | $0$ |
| 60.48.1-30.d.2.3 | $60$ | $2$ | $2$ | $1$ |
| 60.48.1-30.i.2.4 | $60$ | $2$ | $2$ | $1$ |
| 140.192.5-35.a.1.8 | $140$ | $8$ | $8$ | $5$ |
| 140.504.16-35.a.1.2 | $140$ | $21$ | $21$ | $16$ |
| 40.48.1-40.bx.2.1 | $40$ | $2$ | $2$ | $1$ |
| 40.48.1-40.cd.2.2 | $40$ | $2$ | $2$ | $1$ |
| 40.48.1-40.cj.2.5 | $40$ | $2$ | $2$ | $1$ |
| 40.48.1-40.cp.2.4 | $40$ | $2$ | $2$ | $1$ |
| 220.288.9-55.a.1.6 | $220$ | $12$ | $12$ | $9$ |
| 60.48.1-60.k.2.4 | $60$ | $2$ | $2$ | $1$ |
| 60.48.1-60.bd.2.2 | $60$ | $2$ | $2$ | $1$ |
| 260.336.11-65.a.2.5 | $260$ | $14$ | $14$ | $11$ |
| 140.48.1-70.c.2.4 | $140$ | $2$ | $2$ | $1$ |
| 140.48.1-70.d.2.3 | $140$ | $2$ | $2$ | $1$ |
| 220.48.1-110.c.2.2 | $220$ | $2$ | $2$ | $1$ |
| 220.48.1-110.d.2.1 | $220$ | $2$ | $2$ | $1$ |
| 120.48.1-120.en.2.10 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-120.et.2.7 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-120.jl.2.10 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-120.jr.2.3 | $120$ | $2$ | $2$ | $1$ |
| 260.48.1-130.c.2.1 | $260$ | $2$ | $2$ | $1$ |
| 260.48.1-130.d.2.1 | $260$ | $2$ | $2$ | $1$ |
| 140.48.1-140.g.2.6 | $140$ | $2$ | $2$ | $1$ |
| 140.48.1-140.j.2.5 | $140$ | $2$ | $2$ | $1$ |
| 220.48.1-220.g.2.3 | $220$ | $2$ | $2$ | $1$ |
| 220.48.1-220.j.2.2 | $220$ | $2$ | $2$ | $1$ |
| 260.48.1-260.g.2.6 | $260$ | $2$ | $2$ | $1$ |
| 260.48.1-260.j.2.4 | $260$ | $2$ | $2$ | $1$ |
| 280.48.1-280.gm.2.9 | $280$ | $2$ | $2$ | $1$ |
| 280.48.1-280.gp.2.10 | $280$ | $2$ | $2$ | $1$ |
| 280.48.1-280.gy.2.9 | $280$ | $2$ | $2$ | $1$ |
| 280.48.1-280.hb.2.9 | $280$ | $2$ | $2$ | $1$ |