Invariants
| Level: | $20$ | $\SL_2$-level: | $4$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $4^{6}$ | Cusp orbits | $2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 4G0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 20.24.0.16 |
Level structure
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 8 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4\cdot3^3}{5^2}\cdot\frac{(x-8y)^{24}(x^{4}-20x^{3}y+360x^{2}y^{2}-800xy^{3}+1600y^{4})^{3}(7x^{4}-20x^{3}y-120x^{2}y^{2}-800xy^{3}+11200y^{4})^{3}}{(x-8y)^{24}(x^{2}-10xy-20y^{2})^{4}(x^{2}-4xy+40y^{2})^{4}(x^{2}+20xy-80y^{2})^{4}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 4.12.0.a.1 | $4$ | $2$ | $2$ | $0$ | $0$ |
| 20.12.0.b.1 | $20$ | $2$ | $2$ | $0$ | $0$ |
| 20.12.0.o.1 | $20$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 20.120.8.d.1 | $20$ | $5$ | $5$ | $8$ |
| 20.144.7.d.1 | $20$ | $6$ | $6$ | $7$ |
| 20.240.15.d.1 | $20$ | $10$ | $10$ | $15$ |
| 40.48.0.g.1 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0.h.1 | $40$ | $2$ | $2$ | $0$ |
| 40.48.1.j.1 | $40$ | $2$ | $2$ | $1$ |
| 40.48.1.k.1 | $40$ | $2$ | $2$ | $1$ |
| 40.48.1.bp.1 | $40$ | $2$ | $2$ | $1$ |
| 40.48.1.bq.1 | $40$ | $2$ | $2$ | $1$ |
| 40.48.2.b.1 | $40$ | $2$ | $2$ | $2$ |
| 40.48.2.c.1 | $40$ | $2$ | $2$ | $2$ |
| 60.72.4.b.1 | $60$ | $3$ | $3$ | $4$ |
| 60.96.3.b.1 | $60$ | $4$ | $4$ | $3$ |
| 120.48.0.p.1 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0.q.1 | $120$ | $2$ | $2$ | $0$ |
| 120.48.1.bk.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1.bm.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1.eu.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1.ew.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.2.c.1 | $120$ | $2$ | $2$ | $2$ |
| 120.48.2.d.1 | $120$ | $2$ | $2$ | $2$ |
| 140.192.11.b.1 | $140$ | $8$ | $8$ | $11$ |
| 220.288.19.b.1 | $220$ | $12$ | $12$ | $19$ |
| 260.336.23.b.1 | $260$ | $14$ | $14$ | $23$ |
| 280.48.0.g.1 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0.h.1 | $280$ | $2$ | $2$ | $0$ |
| 280.48.1.bo.1 | $280$ | $2$ | $2$ | $1$ |
| 280.48.1.bp.1 | $280$ | $2$ | $2$ | $1$ |
| 280.48.1.cu.1 | $280$ | $2$ | $2$ | $1$ |
| 280.48.1.cv.1 | $280$ | $2$ | $2$ | $1$ |
| 280.48.2.b.1 | $280$ | $2$ | $2$ | $2$ |
| 280.48.2.c.1 | $280$ | $2$ | $2$ | $2$ |