Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | ||||
| Index: | $12$ | $\PSL_2$-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: | 40.12.0.22 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&4\\7&23\end{bmatrix}$, $\begin{bmatrix}19&10\\28&1\end{bmatrix}$, $\begin{bmatrix}21&10\\4&19\end{bmatrix}$, $\begin{bmatrix}39&8\\9&11\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 4.6.0.b.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $24$ |
| Cyclic 40-torsion field degree: | $384$ |
| Full 40-torsion field degree: | $61440$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 11629 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 6 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{6}(x^{2}+48y^{2})^{3}}{y^{4}x^{6}(x^{2}+64y^{2})}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 40.24.0-4.a.1.3 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0-4.c.1.2 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0-8.c.1.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0-8.h.1.1 | $40$ | $2$ | $2$ | $0$ |
| 120.24.0-12.e.1.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-12.f.1.2 | $120$ | $2$ | $2$ | $0$ |
| 120.36.1-12.b.1.16 | $120$ | $3$ | $3$ | $1$ |
| 120.48.0-12.f.1.9 | $120$ | $4$ | $4$ | $0$ |
| 40.24.0-20.e.1.3 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0-20.f.1.4 | $40$ | $2$ | $2$ | $0$ |
| 40.60.2-20.b.1.6 | $40$ | $5$ | $5$ | $2$ |
| 40.72.1-20.b.1.9 | $40$ | $6$ | $6$ | $1$ |
| 40.120.3-20.b.1.3 | $40$ | $10$ | $10$ | $3$ |
| 120.24.0-24.m.1.2 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-24.p.1.2 | $120$ | $2$ | $2$ | $0$ |
| 280.24.0-28.e.1.3 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0-28.f.1.4 | $280$ | $2$ | $2$ | $0$ |
| 280.96.2-28.b.1.15 | $280$ | $8$ | $8$ | $2$ |
| 280.252.7-28.b.1.6 | $280$ | $21$ | $21$ | $7$ |
| 280.336.9-28.b.1.9 | $280$ | $28$ | $28$ | $9$ |
| 40.24.0-40.m.1.4 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0-40.p.1.4 | $40$ | $2$ | $2$ | $0$ |
| 280.24.0-56.m.1.3 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0-56.p.1.3 | $280$ | $2$ | $2$ | $0$ |
| 120.24.0-60.e.1.6 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-60.f.1.8 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-120.m.1.4 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-120.p.1.4 | $120$ | $2$ | $2$ | $0$ |
| 280.24.0-140.e.1.4 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0-140.f.1.2 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0-280.m.1.6 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0-280.p.1.6 | $280$ | $2$ | $2$ | $0$ |