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: | 40.48.0.825 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}7&12\\15&31\end{bmatrix}$, $\begin{bmatrix}7&32\\26&33\end{bmatrix}$, $\begin{bmatrix}31&28\\25&13\end{bmatrix}$, $\begin{bmatrix}33&24\\12&3\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.24.0.bs.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 16 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^2\cdot3^3}{5}\cdot\frac{(x-4y)^{24}(107x^{8}+2960x^{7}y+34400x^{6}y^{2}+243200x^{5}y^{3}+1107200x^{4}y^{4}+2048000x^{3}y^{5}+5120000x^{2}y^{6}-40960000xy^{7}+81920000y^{8})^{3}}{(x-4y)^{24}(x^{2}-20xy-80y^{2})^{8}(x^{2}+4xy+40y^{2})^{2}(x^{2}+10xy-20y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.24.0-8.o.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0-20.g.1.2 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0-20.g.1.3 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0-8.o.1.4 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0-40.z.1.11 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0-40.z.1.16 | $40$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.