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 | $4^{8}\cdot8^{2}$ | Cusp orbits | $2^{3}\cdot4$ | ||
| 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: | 8N0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.96.0.43 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}17&30\\32&7\end{bmatrix}$, $\begin{bmatrix}29&24\\0&17\end{bmatrix}$, $\begin{bmatrix}31&22\\20&11\end{bmatrix}$, $\begin{bmatrix}33&26\\36&39\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.48.0.n.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 2 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^2}{3^{12}\cdot5^2}\cdot\frac{(9x-25y)^{48}(240820356763374108201x^{16}-5148829897937690274864x^{15}y+51602645423712081612600x^{14}y^{2}-321808034882581268933520x^{13}y^{3}+1397766681051634655451900x^{12}y^{4}-4484146959736415240964912x^{11}y^{5}+10993155457911933599943048x^{10}y^{6}-21016610356834766064675600x^{9}y^{7}+31688721882277240500406710x^{8}y^{8}-37856780750721579602000400x^{7}y^{9}+35789117021773001384947848x^{6}y^{10}-26581063686899227149156912x^{5}y^{11}+15278383026448174985442300x^{4}y^{12}-6612654279371692472741520x^{3}y^{13}+2048128254015895415878200x^{2}y^{14}-408725433970868259407664xy^{15}+39824488795497854053801y^{16})^{3}}{(9x-25y)^{48}(9x^{2}-22xy+9y^{2})^{4}(81x^{2}-558xy+601y^{2})^{8}(81x^{2}-234xy+205y^{2})^{4}(124659x^{4}-667764x^{3}y+1350594x^{2}y^{2}-1248084xy^{3}+479299y^{4})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.48.0-8.d.2.16 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-8.d.2.6 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-40.e.1.6 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-40.e.1.7 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-40.i.1.12 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-40.i.1.30 | $40$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.