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: | 48.96.0.1455 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}11&32\\28&29\end{bmatrix}$, $\begin{bmatrix}13&13\\4&33\end{bmatrix}$, $\begin{bmatrix}17&33\\32&35\end{bmatrix}$, $\begin{bmatrix}27&8\\40&3\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.0.y.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 6 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3}\cdot\frac{(x+y)^{48}(x^{16}+16x^{15}y+864x^{14}y^{2}+10976x^{13}y^{3}+102032x^{12}y^{4}+665280x^{11}y^{5}+3332224x^{10}y^{6}+12992128x^{9}y^{7}+39945312x^{8}y^{8}+97018624x^{7}y^{9}+185532928x^{6}y^{10}+276466176x^{5}y^{11}+314730752x^{4}y^{12}+264578048x^{3}y^{13}+154884096x^{2}y^{14}+56424448xy^{15}+9634048y^{16})^{3}}{y^{2}(x+y)^{50}(x^{2}+2xy-2y^{2})^{16}(x^{2}+2xy+4y^{2})^{2}(x^{4}+4x^{3}y+24x^{2}y^{2}+40xy^{3}+28y^{4})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 16.48.0-16.e.2.3 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-24.bl.1.6 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-16.e.2.16 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.e.2.9 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.e.2.16 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-24.bl.1.3 | $48$ | $2$ | $2$ | $0$ | $0$ |
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.cv.1.1 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.cw.1.5 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.dd.1.5 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.de.1.1 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.eb.1.1 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.ec.1.6 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.ej.1.3 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.ek.1.1 | $48$ | $2$ | $2$ | $1$ |
| 48.288.8-48.dk.2.5 | $48$ | $3$ | $3$ | $8$ |
| 48.384.7-48.ej.2.9 | $48$ | $4$ | $4$ | $7$ |
| 240.192.1-240.mp.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.mq.1.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.mx.1.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.my.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.rn.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.ro.1.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.rv.1.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.rw.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.480.16-240.cc.2.14 | $240$ | $5$ | $5$ | $16$ |