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 | $2^{8}\cdot16^{2}$ | 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: | 16G0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.96.0.1342 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&45\\24&31\end{bmatrix}$, $\begin{bmatrix}5&27\\24&31\end{bmatrix}$, $\begin{bmatrix}17&31\\12&13\end{bmatrix}$, $\begin{bmatrix}23&31\\28&7\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.0.x.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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2}{3}\cdot\frac{(4x-5y)^{48}(55316511195136x^{16}+32860123693056x^{15}y-324897582612480x^{14}y^{2}+53318579650560x^{13}y^{3}+7103427426385920x^{12}y^{4}-43652866482634752x^{11}y^{5}+157866290010390528x^{10}y^{6}-410846699932876800x^{9}y^{7}+820083680460693504x^{8}y^{8}-1283508134903513088x^{7}y^{9}+1579283234716078080x^{6}y^{10}-1513842938653667328x^{5}y^{11}+1108464152443119360x^{4}y^{12}-599362477874689536x^{3}y^{13}+225701255001339072x^{2}y^{14}-52883953046615232xy^{15}+5808381433436961y^{16})^{3}}{(4x-5y)^{48}(8x^{2}+36xy-33y^{2})^{2}(24x^{2}-32xy+27y^{2})^{2}(40x^{2}-72xy+3y^{2})^{16}(1856x^{4}-3456x^{3}y+8496x^{2}y^{2}-9936xy^{3}+4365y^{4})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 16.48.0-16.f.1.2 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-24.bj.1.12 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-16.f.1.4 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.f.1.4 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.f.1.17 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-24.bj.1.6 | $48$ | $2$ | $2$ | $0$ | $0$ |
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.cl.2.3 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.cm.1.3 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.ct.2.3 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.cu.1.5 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.dr.2.3 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.ds.2.3 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.dz.2.5 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.ea.2.3 | $48$ | $2$ | $2$ | $1$ |
| 48.288.8-48.dj.2.1 | $48$ | $3$ | $3$ | $8$ |
| 48.384.7-48.eg.2.21 | $48$ | $4$ | $4$ | $7$ |
| 240.192.1-240.mf.2.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.mg.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.mn.2.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.mo.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.rd.2.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.re.2.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.rl.2.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.rm.2.5 | $240$ | $2$ | $2$ | $1$ |
| 240.480.16-240.cb.2.4 | $240$ | $5$ | $5$ | $16$ |