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 | $2^{4}\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| 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: | 8G0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.48.0.274 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}11&28\\24&15\end{bmatrix}$, $\begin{bmatrix}25&12\\14&5\end{bmatrix}$, $\begin{bmatrix}27&12\\10&29\end{bmatrix}$, $\begin{bmatrix}35&4\\28&29\end{bmatrix}$, $\begin{bmatrix}35&8\\2&5\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.24.0.m.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 44 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^2}{5^4}\cdot\frac{(10x-3y)^{24}(96160000x^{8}-242560000x^{7}y+254560000x^{6}y^{2}-148768000x^{5}y^{3}+57141600x^{4}y^{4}-13524800x^{3}y^{5}+2143600x^{2}y^{6}-154160xy^{7}+5161y^{8})^{3}}{(2x+y)^{8}(10x-3y)^{32}(20x^{2}-20xy+y^{2})^{2}(60x^{2}-20xy+7y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.24.0-4.b.1.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0-4.b.1.2 | $40$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.