Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | ||||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (of which $2$ are rational) | Cusp widths | $4^{8}\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}\cdot4$ | ||
| 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: | 8N0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.96.0.891 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&8\\24&11\end{bmatrix}$, $\begin{bmatrix}17&4\\0&1\end{bmatrix}$, $\begin{bmatrix}23&4\\28&31\end{bmatrix}$, $\begin{bmatrix}23&32\\30&19\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.48.0.e.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $192$ |
| Full 40-torsion field degree: | $7680$ |
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 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^2}\cdot\frac{(x+y)^{48}(83233x^{16}+228528x^{15}y+294904x^{14}y^{2}+246288x^{13}y^{3}+215964x^{12}y^{4}+488880x^{11}y^{5}+1490888x^{10}y^{6}+3214224x^{9}y^{7}+4251398x^{8}y^{8}+3214224x^{7}y^{9}+1490888x^{6}y^{10}+488880x^{5}y^{11}+215964x^{4}y^{12}+246288x^{3}y^{13}+294904x^{2}y^{14}+228528xy^{15}+83233y^{16})^{3}}{(x-y)^{4}(x+y)^{52}(x^{2}+6xy+y^{2})^{8}(3x^{2}+2xy+3y^{2})^{4}(17x^{4}+12x^{3}y+6x^{2}y^{2}+12xy^{3}+17y^{4})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.48.0-8.c.1.3 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-8.c.1.4 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-8.d.1.11 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-8.d.1.13 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-8.e.1.2 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-8.e.1.5 | $40$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.