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 | $1^{2}\cdot2^{4}$ | ||
| 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: | 40.96.0.568 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}7&12\\8&25\end{bmatrix}$, $\begin{bmatrix}25&4\\26&13\end{bmatrix}$, $\begin{bmatrix}31&20\\10&29\end{bmatrix}$, $\begin{bmatrix}39&20\\18&3\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.48.0.f.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| 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 7 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^2}\cdot\frac{(x-y)^{48}(x^{16}-32x^{15}y+464x^{14}y^{2}-4032x^{13}y^{3}+24304x^{12}y^{4}-117376x^{11}y^{5}+514752x^{10}y^{6}-2061056x^{9}y^{7}+6858848x^{8}y^{8}-17491456x^{7}y^{9}+32881408x^{6}y^{10}-44921856x^{5}y^{11}+44543744x^{4}y^{12}-32356352x^{3}y^{13}+17347584x^{2}y^{14}-6492160xy^{15}+1278208y^{16})^{3}}{y^{4}(x-2y)^{4}(x-y)^{48}(x^{2}-2y^{2})^{4}(x^{2}-8xy+14y^{2})^{4}(x^{2}-4xy+2y^{2})^{4}(x^{2}-4xy+6y^{2})^{8}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.48.0-8.c.1.6 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-8.c.1.9 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-8.e.1.6 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-8.e.1.9 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-8.e.2.2 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-8.e.2.16 | $40$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.