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 | $2^{4}\cdot4^{2}\cdot8^{4}$ | 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: | 8O0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.96.0.1068 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}15&5\\4&39\end{bmatrix}$, $\begin{bmatrix}19&22\\36&5\end{bmatrix}$, $\begin{bmatrix}29&34\\4&3\end{bmatrix}$, $\begin{bmatrix}45&32\\4&23\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.48.0.bl.2 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2\cdot3\cdot5^4}\cdot\frac{(2x+3y)^{48}(317399296x^{16}-5054447616x^{15}y+28845726720x^{14}y^{2}+10996899840x^{13}y^{3}+92799717120x^{12}y^{4}+329187151872x^{11}y^{5}+17498827008x^{10}y^{6}-393885296640x^{9}y^{7}-256668043680x^{8}y^{8}+590827944960x^{7}y^{9}+39372360768x^{6}y^{10}-1111006637568x^{5}y^{11}+469798567920x^{4}y^{12}-83507708160x^{3}y^{13}+328570855920x^{2}y^{14}+86359976064xy^{15}+8134596801y^{16})^{3}}{(x-y)^{2}(2x+3y)^{50}(2x^{2}+3y^{2})^{4}(2x^{2}-24xy-3y^{2})^{2}(92x^{4}+192x^{3}y-828x^{2}y^{2}-288xy^{3}+207y^{4})^{8}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 16.48.0-8.bb.2.7 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-8.bb.2.7 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-24.bj.1.1 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-24.bj.1.6 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-24.by.2.8 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-24.by.2.15 | $48$ | $2$ | $2$ | $0$ | $0$ |
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.ck.1.3 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.cm.1.1 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.cs.2.2 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.cu.2.2 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.dq.1.4 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.ds.1.1 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.dy.2.1 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.ea.2.3 | $48$ | $2$ | $2$ | $1$ |
| 48.288.8-24.gn.2.6 | $48$ | $3$ | $3$ | $8$ |
| 48.384.7-24.em.1.1 | $48$ | $4$ | $4$ | $7$ |
| 240.192.1-240.mc.1.3 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.me.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.mk.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.mm.2.3 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.ra.1.3 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.rc.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.ri.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.rk.2.3 | $240$ | $2$ | $2$ | $1$ |
| 240.480.16-120.ev.1.15 | $240$ | $5$ | $5$ | $16$ |