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: | 48.96.0.1312 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&33\\8&19\end{bmatrix}$, $\begin{bmatrix}9&47\\4&33\end{bmatrix}$, $\begin{bmatrix}19&1\\20&19\end{bmatrix}$, $\begin{bmatrix}23&22\\24&35\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.0.bk.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 5 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^2\cdot3^2\cdot5^2}\cdot\frac{(2x+3y)^{48}(1356595456x^{16}+5544321024x^{15}y-8580910080x^{14}y^{2}-36328181760x^{13}y^{3}+3189784320x^{12}y^{4}+22230484992x^{11}y^{5}+38009426688x^{10}y^{6}+80929704960x^{9}y^{7}+22826279520x^{8}y^{8}-121394557440x^{7}y^{9}+85521210048x^{6}y^{10}-75027886848x^{5}y^{11}+16148283120x^{4}y^{12}+275867130240x^{3}y^{13}-97741928880x^{2}y^{14}-94729922496xy^{15}+34768057761y^{16})^{3}}{(x-y)^{4}(2x+3y)^{52}(2x^{2}+3y^{2})^{2}(2x^{2}-24xy-3y^{2})^{16}(92x^{4}+192x^{3}y-828x^{2}y^{2}-288xy^{3}+207y^{4})}$ |
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.by.1.13 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-16.e.1.6 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.h.1.19 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.h.1.29 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-24.by.1.5 | $48$ | $2$ | $2$ | $0$ | $0$ |
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.m.2.7 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.y.1.3 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.bf.2.5 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.bz.2.2 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.cf.2.3 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.cs.1.5 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.cw.2.3 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.dh.2.3 | $48$ | $2$ | $2$ | $1$ |
| 48.288.8-48.ic.1.5 | $48$ | $3$ | $3$ | $8$ |
| 48.384.7-48.gx.1.21 | $48$ | $4$ | $4$ | $7$ |
| 240.192.1-240.xb.2.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.xj.1.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.yh.2.3 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.yp.2.2 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.zn.2.3 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.zv.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bat.2.3 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bbb.2.2 | $240$ | $2$ | $2$ | $1$ |
| 240.480.16-240.fw.2.4 | $240$ | $5$ | $5$ | $16$ |