Invariants
| Level: | $40$ | $\SL_2$-level: | $4$ | ||||
| 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 | $4^{6}$ | 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: | 4G0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.48.0.315 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}9&22\\28&31\end{bmatrix}$, $\begin{bmatrix}15&16\\34&29\end{bmatrix}$, $\begin{bmatrix}27&4\\28&23\end{bmatrix}$, $\begin{bmatrix}29&6\\6&17\end{bmatrix}$, $\begin{bmatrix}37&32\\38&11\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 20.24.0.c.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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{5^2}\cdot\frac{(10x-3y)^{24}(9760000x^{8}-12160000x^{7}y+14560000x^{6}y^{2}-10528000x^{5}y^{3}+3813600x^{4}y^{4}-660800x^{3}y^{5}+103600x^{2}y^{6}-19760xy^{7}+2221y^{8})^{3}}{(2x+y)^{4}(10x-3y)^{28}(20x^{2}-20xy+y^{2})^{4}(60x^{2}-20xy+7y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.24.0-4.b.1.6 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0-4.b.1.1 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0-20.a.1.1 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0-20.a.1.3 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0-20.b.1.2 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0-20.b.1.3 | $40$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.