Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $2^{2}$ | ||
| 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: | 4E0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.24.0.395 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}15&6\\3&31\end{bmatrix}$, $\begin{bmatrix}29&14\\36&13\end{bmatrix}$, $\begin{bmatrix}31&32\\37&33\end{bmatrix}$, $\begin{bmatrix}37&18\\6&13\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 20.12.0.d.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $24$ |
| Cyclic 40-torsion field degree: | $384$ |
| Full 40-torsion field degree: | $30720$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 150 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{5^2}\cdot\frac{(x+2y)^{12}(11x^{4}+32x^{3}y-26x^{2}y^{2}-32xy^{3}+11y^{4})^{3}}{(x+2y)^{12}(x^{2}+y^{2})^{4}(x^{2}+xy-y^{2})^{2}}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.