Invariants
| Level: | $48$ | $\SL_2$-level: | $16$ | ||||
| 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 | $2^{4}\cdot4^{2}\cdot8^{4}$ | 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: | 8O0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.96.0.657 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&21\\40&7\end{bmatrix}$, $\begin{bmatrix}3&29\\32&25\end{bmatrix}$, $\begin{bmatrix}5&3\\0&11\end{bmatrix}$, $\begin{bmatrix}11&38\\32&19\end{bmatrix}$, $\begin{bmatrix}11&40\\16&7\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.48.0.be.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $8$ |
| Cyclic 48-torsion field degree: | $64$ |
| Full 48-torsion field degree: | $12288$ |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^6\cdot3^2}\cdot\frac{x^{48}(81x^{8}-3456x^{6}y^{2}+4608x^{4}y^{4}+98304x^{2}y^{6}+65536y^{8})^{3}(81x^{8}+3456x^{6}y^{2}+4608x^{4}y^{4}-98304x^{2}y^{6}+65536y^{8})^{3}}{y^{4}x^{52}(3x^{2}-16y^{2})^{2}(3x^{2}+16y^{2})^{2}(9x^{4}+256y^{4})^{8}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 16.48.0-8.q.1.2 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-8.q.1.3 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-24.by.1.4 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-24.by.1.5 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-24.bz.2.4 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-24.bz.2.5 | $48$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.