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 | $4^{6}$ | 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: | 40.48.0.419 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}11&2\\12&31\end{bmatrix}$, $\begin{bmatrix}31&32\\38&37\end{bmatrix}$, $\begin{bmatrix}33&34\\12&17\end{bmatrix}$, $\begin{bmatrix}33&34\\32&5\end{bmatrix}$, $\begin{bmatrix}35&18\\16&3\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 20.24.0.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: | $15360$ |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4\cdot3^3}{5^2}\cdot\frac{(x-8y)^{24}(x^{4}-20x^{3}y+360x^{2}y^{2}-800xy^{3}+1600y^{4})^{3}(7x^{4}-20x^{3}y-120x^{2}y^{2}-800xy^{3}+11200y^{4})^{3}}{(x-8y)^{24}(x^{2}-10xy-20y^{2})^{4}(x^{2}-4xy+40y^{2})^{4}(x^{2}+20xy-80y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.24.0-4.a.1.5 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0-4.a.1.2 | $40$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.