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 | $2^{8}\cdot16^{2}$ | 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: | 16G0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.96.0.356 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}7&26\\24&47\end{bmatrix}$, $\begin{bmatrix}11&30\\24&17\end{bmatrix}$, $\begin{bmatrix}13&24\\32&25\end{bmatrix}$, $\begin{bmatrix}13&30\\32&23\end{bmatrix}$, $\begin{bmatrix}35&22\\8&27\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.0.d.2 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^8}\cdot\frac{(x+y)^{48}(6481x^{16}-1264x^{15}y-8592x^{14}y^{2}-31808x^{13}y^{3}-61600x^{12}y^{4}-5376x^{11}y^{5}+367360x^{10}y^{6}+1381376x^{9}y^{7}+3273984x^{8}y^{8}+5857280x^{7}y^{9}+8200192x^{6}y^{10}+8945664x^{5}y^{11}+7454720x^{4}y^{12}+4587520x^{3}y^{13}+1966080x^{2}y^{14}+524288xy^{15}+65536y^{16})^{3}}{x^{16}(x+y)^{48}(x+2y)^{16}(x^{2}-2xy-2y^{2})^{2}(x^{2}+xy+y^{2})^{2}(5x^{4}+4x^{3}y+12x^{2}y^{2}+16xy^{3}+8y^{4})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 16.48.0-8.i.1.2 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-8.i.1.2 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.e.2.9 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.e.2.24 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.f.1.1 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.f.1.32 | $48$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.