Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $4^{6}$ | Cusp orbits | $1^{2}\cdot4$ | ||
| 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: | 4G0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.48.0.571 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}7&16\\26&7\end{bmatrix}$, $\begin{bmatrix}11&16\\1&33\end{bmatrix}$, $\begin{bmatrix}17&12\\37&19\end{bmatrix}$, $\begin{bmatrix}33&32\\19&7\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 20.24.0.d.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $192$ |
| Full 40-torsion field degree: | $15360$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 19 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^{10}\cdot5^2}\cdot\frac{x^{24}(25x^{4}-2560x^{2}y^{2}+16384y^{4})^{3}(25x^{4}+2560x^{2}y^{2}+16384y^{4})^{3}}{y^{4}x^{28}(25x^{4}+16384y^{4})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.24.0-4.d.1.5 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0-4.d.1.2 | $40$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.