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 | $1^{4}\cdot2^{2}\cdot4^{2}\cdot16^{2}$ | 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: | 16H0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.96.0.1309 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&33\\8&11\end{bmatrix}$, $\begin{bmatrix}17&30\\20&11\end{bmatrix}$, $\begin{bmatrix}35&32\\24&19\end{bmatrix}$, $\begin{bmatrix}37&17\\12&13\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.0.bl.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 4 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^2\cdot3^2}\cdot\frac{(3x+y)^{48}(6561x^{16}+34992x^{14}y^{2}-664848x^{12}y^{4}-1881792x^{10}y^{6}+3408480x^{8}y^{8}-836352x^{6}y^{10}-131328x^{4}y^{12}+3072x^{2}y^{14}+256y^{16})^{3}}{y^{4}x^{4}(3x+y)^{48}(3x^{2}-2y^{2})^{2}(3x^{2}+2y^{2})^{16}(9x^{4}+36x^{2}y^{2}+4y^{4})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 16.48.0-16.e.1.5 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-24.bz.2.5 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-16.e.1.5 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.h.1.3 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.h.1.4 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-24.bz.2.13 | $48$ | $2$ | $2$ | $0$ | $0$ |
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.o.1.7 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.y.2.2 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.bf.2.3 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.ca.1.1 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.cg.1.5 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.cr.2.3 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.cv.2.1 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.di.2.1 | $48$ | $2$ | $2$ | $1$ |
| 48.288.8-48.id.1.19 | $48$ | $3$ | $3$ | $8$ |
| 48.384.7-48.gy.2.9 | $48$ | $4$ | $4$ | $7$ |
| 240.192.1-240.xc.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.xk.2.3 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.yi.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.yq.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.zo.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.zw.2.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bau.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bbc.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.480.16-240.fx.2.2 | $240$ | $5$ | $5$ | $16$ |