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: | 48.96.0.971 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}9&16\\44&11\end{bmatrix}$, $\begin{bmatrix}15&43\\16&21\end{bmatrix}$, $\begin{bmatrix}17&1\\0&31\end{bmatrix}$, $\begin{bmatrix}37&0\\36&19\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.0.u.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 7 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^8}{3}\cdot\frac{(3x+y)^{48}(81x^{8}+648x^{7}y+1512x^{6}y^{2}+1512x^{5}y^{3}+630x^{4}y^{4}-84x^{2}y^{6}-24xy^{7}-2y^{8})^{3}(162x^{8}+648x^{7}y+756x^{6}y^{2}-630x^{4}y^{4}-504x^{3}y^{5}-168x^{2}y^{6}-24xy^{7}-y^{8})^{3}}{(x+y)^{2}(3x+y)^{50}(3x^{2}-y^{2})^{2}(3x^{2}+3xy+y^{2})^{16}(9x^{4}+36x^{3}y+36x^{2}y^{2}+12xy^{3}+y^{4})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 16.48.0-16.e.2.1 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-24.bh.1.2 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-16.e.2.16 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.f.1.1 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.f.1.24 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-24.bh.1.1 | $48$ | $2$ | $2$ | $0$ | $0$ |
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.cf.1.1 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.cg.2.1 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.cn.1.1 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.co.1.2 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.dl.1.1 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.dm.1.2 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.dt.1.1 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.du.2.1 | $48$ | $2$ | $2$ | $1$ |
| 48.288.8-48.dc.2.2 | $48$ | $3$ | $3$ | $8$ |
| 48.384.7-48.eb.2.9 | $48$ | $4$ | $4$ | $7$ |
| 240.192.1-240.lj.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.lk.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.lr.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.ls.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.qh.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.qi.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.qp.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.qq.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.480.16-240.bu.2.2 | $240$ | $5$ | $5$ | $16$ |