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: | $10$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $2^{5}$ | ||
| 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: | 8O0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.96.0.183 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}9&32\\34&29\end{bmatrix}$, $\begin{bmatrix}21&24\\6&11\end{bmatrix}$, $\begin{bmatrix}25&24\\38&19\end{bmatrix}$, $\begin{bmatrix}33&24\\32&37\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.48.0.bd.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $6$ |
| Cyclic 40-torsion field degree: | $96$ |
| 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 3 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^2\cdot5^2}\cdot\frac{(4x+y)^{48}(4043309056x^{16}+43956305920x^{15}y+182452224000x^{14}y^{2}+348127232000x^{13}y^{3}+401997824000x^{12}y^{4}+149422080000x^{11}y^{5}-152453120000x^{10}y^{6}-167567360000x^{9}y^{7}-97896960000x^{8}y^{8}-104729600000x^{7}y^{9}-59552000000x^{6}y^{10}+36480000000x^{5}y^{11}+61340000000x^{4}y^{12}+33200000000x^{3}y^{13}+10875000000x^{2}y^{14}+1637500000xy^{15}+94140625y^{16})^{3}}{(4x+y)^{48}(4x^{2}-10xy-5y^{2})^{8}(8x^{2}-5y^{2})^{2}(8x^{2}+4xy+5y^{2})^{4}(8x^{2}+40xy+5y^{2})^{2}(16x^{2}+20xy-5y^{2})^{8}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.48.0-8.i.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-8.i.1.11 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-40.i.1.3 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-40.i.1.13 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-40.cb.2.3 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-40.cb.2.14 | $40$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.