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: | 48.96.0.843 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}9&8\\4&11\end{bmatrix}$, $\begin{bmatrix}17&17\\28&33\end{bmatrix}$, $\begin{bmatrix}23&44\\36&37\end{bmatrix}$, $\begin{bmatrix}33&17\\20&13\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.0.bp.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $8$ |
| Cyclic 48-torsion field degree: | $32$ |
| 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^{24}\cdot3^8}\cdot\frac{x^{48}(6561x^{16}-139968x^{14}y^{2}+559872x^{12}y^{4}+995328x^{10}y^{6}-3317760x^{8}y^{8}+7077888x^{6}y^{10}+28311552x^{4}y^{12}-50331648x^{2}y^{14}+16777216y^{16})^{3}}{y^{16}x^{64}(3x^{2}-8y^{2})^{4}(3x^{2}+8y^{2})^{2}(9x^{4}-144x^{2}y^{2}+64y^{4})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 16.48.0-8.bb.1.8 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-8.bb.1.6 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.f.1.1 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.f.1.11 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.h.1.1 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.h.1.3 | $48$ | $2$ | $2$ | $0$ | $0$ |
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.f.2.2 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.ba.1.1 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.bm.2.1 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.by.1.1 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.cg.2.1 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.cu.1.1 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.cy.2.1 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.di.2.1 | $48$ | $2$ | $2$ | $1$ |
| 48.288.8-48.ip.1.17 | $48$ | $3$ | $3$ | $8$ |
| 48.384.7-48.hc.1.2 | $48$ | $4$ | $4$ | $7$ |
| 240.192.1-240.xg.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.xo.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.ym.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.yu.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.zs.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.baa.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bay.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bbg.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.480.16-240.gj.1.2 | $240$ | $5$ | $5$ | $16$ |