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 | $2^{4}\cdot4^{2}\cdot8^{4}$ | 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: | 8O0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.96.0.1764 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}11&0\\24&19\end{bmatrix}$, $\begin{bmatrix}25&19\\32&21\end{bmatrix}$, $\begin{bmatrix}29&38\\16&3\end{bmatrix}$, $\begin{bmatrix}37&41\\16&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.48.0.bf.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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2}{3^2}\cdot\frac{x^{48}(75563008x^{16}+753401856x^{15}y+3181510656x^{14}y^{2}+7442399232x^{13}y^{3}+10346545152x^{12}y^{4}+7721091072x^{11}y^{5}+530178048x^{10}y^{6}-5538668544x^{9}y^{7}-6928685568x^{8}y^{8}-4154001408x^{7}y^{9}+298225152x^{6}y^{10}+3257335296x^{5}y^{11}+3273711552x^{4}y^{12}+1766116224x^{3}y^{13}+566240544x^{2}y^{14}+100567008xy^{15}+7564833y^{16})^{3}}{x^{48}(4x^{2}-3y^{2})^{2}(4x^{2}+4xy+3y^{2})^{4}(4x^{2}+12xy+3y^{2})^{2}(16x^{4}-96x^{3}y-216x^{2}y^{2}-72xy^{3}+9y^{4})^{8}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 16.48.0-8.r.1.3 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-8.r.1.2 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-24.by.1.2 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-24.by.1.5 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-24.by.2.4 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-24.by.2.7 | $48$ | $2$ | $2$ | $0$ | $0$ |
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.br.1.3 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.br.2.2 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.bv.1.2 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.bv.2.2 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.bz.1.3 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.bz.2.2 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.cb.1.5 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.cb.2.3 | $48$ | $2$ | $2$ | $1$ |
| 48.288.8-24.fz.1.16 | $48$ | $3$ | $3$ | $8$ |
| 48.384.7-24.ec.1.6 | $48$ | $4$ | $4$ | $7$ |
| 240.192.1-240.fd.1.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.fd.2.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.fj.1.3 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.fj.2.3 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.fr.1.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.fr.2.3 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.fx.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.fx.2.2 | $240$ | $2$ | $2$ | $1$ |
| 240.480.16-120.ei.1.5 | $240$ | $5$ | $5$ | $16$ |