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: | $10$ (none of which are rational) | Cusp widths | $2^{8}\cdot16^{2}$ | 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: | 16G0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.96.0.1714 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}5&14\\44&7\end{bmatrix}$, $\begin{bmatrix}13&24\\20&47\end{bmatrix}$, $\begin{bmatrix}41&39\\32&7\end{bmatrix}$, $\begin{bmatrix}43&35\\44&35\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.0.f.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 3 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2}{3^8}\cdot\frac{(4x-3y)^{48}(4096x^{8}-114688x^{7}y+821248x^{6}y^{2}-2627584x^{5}y^{3}+4465024x^{4}y^{4}-4239616x^{3}y^{5}+2181664x^{2}y^{6}-523936xy^{7}+33793y^{8})^{3}(192512x^{8}-1359872x^{7}y+3897344x^{6}y^{2}-5715968x^{5}y^{3}+4290176x^{4}y^{4}-1119488x^{3}y^{5}-519712x^{2}y^{6}+411616xy^{7}-76561y^{8})^{3}}{(4x-3y)^{48}(8x^{2}-20xy+11y^{2})^{2}(8x^{2}-16xy+9y^{2})^{16}(8x^{2}-8xy-y^{2})^{2}(64x^{4}-512x^{3}y+1104x^{2}y^{2}-896xy^{3}+241y^{4})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 16.48.0-8.k.1.3 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-8.k.1.3 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.f.1.1 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.f.1.28 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.f.2.9 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.f.2.20 | $48$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 48.192.1-48.t.1.1 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.t.2.1 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.w.1.1 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.w.2.1 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.ba.1.1 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.ba.2.1 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.bd.1.1 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.bd.2.1 | $48$ | $2$ | $2$ | $1$ |
| 48.288.8-48.w.1.10 | $48$ | $3$ | $3$ | $8$ |
| 48.384.7-48.co.1.17 | $48$ | $4$ | $4$ | $7$ |
| 240.192.1-240.cl.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.cl.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.cs.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.cs.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.da.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.da.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.df.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.df.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.480.16-240.n.1.13 | $240$ | $5$ | $5$ | $16$ |