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: | $6$ (none of which are rational) | Cusp widths | $2^{4}\cdot8^{2}$ | 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: | 8G0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.48.0.55 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}7&4\\6&11\end{bmatrix}$, $\begin{bmatrix}27&12\\37&13\end{bmatrix}$, $\begin{bmatrix}31&0\\37&21\end{bmatrix}$, $\begin{bmatrix}35&16\\1&29\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.24.0.w.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 8 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^2\cdot3^3}{5^4}\cdot\frac{(5x+2y)^{24}(100x^{4}-200x^{3}y-240x^{2}y^{2}-20xy^{3}+y^{4})^{3}(1300x^{4}-200x^{3}y-480x^{2}y^{2}-20xy^{3}+13y^{4})^{3}}{(5x+2y)^{24}(5x^{2}-5xy-y^{2})^{2}(10x^{2}+2xy+y^{2})^{8}(20x^{2}+10xy-y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 4.24.0-4.d.1.2 | $4$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0-4.d.1.6 | $40$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.