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: | 48.96.0.255 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}7&16\\12&37\end{bmatrix}$, $\begin{bmatrix}23&47\\12&43\end{bmatrix}$, $\begin{bmatrix}31&31\\40&33\end{bmatrix}$, $\begin{bmatrix}33&5\\40&3\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.0.bf.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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^{16}\cdot3^8}\cdot\frac{(3x+y)^{48}(6561x^{16}+69984x^{14}y^{2}+139968x^{12}y^{4}-124416x^{10}y^{6}-207360x^{8}y^{8}-221184x^{6}y^{10}+442368x^{4}y^{12}+393216x^{2}y^{14}+65536y^{16})^{3}}{y^{16}x^{16}(3x+y)^{48}(3x^{2}-4y^{2})^{2}(3x^{2}+4y^{2})^{4}(9x^{4}+72x^{2}y^{2}+16y^{4})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.48.0-8.bb.1.7 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-8.bb.1.2 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.e.1.19 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.e.1.29 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.g.1.29 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.g.1.31 | $48$ | $2$ | $2$ | $0$ | $0$ |
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.i.2.12 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.v.2.5 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.bj.2.7 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.bu.2.7 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.ck.2.6 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.cq.2.8 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.dc.2.8 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.de.2.8 | $48$ | $2$ | $2$ | $1$ |
| 48.288.8-48.hp.2.1 | $48$ | $3$ | $3$ | $8$ |
| 48.384.7-48.gs.1.28 | $48$ | $4$ | $4$ | $7$ |
| 240.192.1-240.wo.2.14 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.ww.2.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.xu.2.8 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.yc.2.10 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.za.2.11 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.zi.2.14 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bag.2.14 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bao.2.8 | $240$ | $2$ | $2$ | $1$ |
| 240.480.16-240.fj.1.12 | $240$ | $5$ | $5$ | $16$ |