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 | $1^{4}\cdot2^{2}\cdot4^{2}\cdot16^{2}$ | 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: | 16H0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.96.0.154 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}11&0\\28&29\end{bmatrix}$, $\begin{bmatrix}11&21\\28&43\end{bmatrix}$, $\begin{bmatrix}19&1\\20&3\end{bmatrix}$, $\begin{bmatrix}29&43\\0&11\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.0.bg.2 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $8$ |
| Cyclic 48-torsion field degree: | $128$ |
| 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2}{3^8}\cdot\frac{(3x-4y)^{48}(3336655599x^{16}-35010545760x^{15}y+91616684256x^{14}y^{2}+518266512000x^{13}y^{3}-5176551457728x^{12}y^{4}+21531215854080x^{11}y^{5}-56700226727424x^{10}y^{6}+104707000166400x^{9}y^{7}-140894854202880x^{8}y^{8}+139609333555200x^{7}y^{9}-100800403070976x^{6}y^{10}+51036956098560x^{5}y^{11}-16360458928128x^{4}y^{12}+2183970816000x^{3}y^{13}+514762604544x^{2}y^{14}-262282936320xy^{15}+33328922624y^{16})^{3}}{(3x-4y)^{48}(3x^{2}-4y^{2})^{2}(9x^{2}-20xy+12y^{2})^{16}(15x^{2}-36xy+20y^{2})^{4}(117x^{4}-540x^{3}y+936x^{2}y^{2}-720xy^{3}+208y^{4})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.48.0-8.ba.2.6 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-8.ba.2.6 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.f.1.23 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.f.1.25 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.g.1.15 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.g.1.29 | $48$ | $2$ | $2$ | $0$ | $0$ |
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.g.1.4 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.w.2.4 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.bn.2.3 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.bt.1.4 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.cl.2.4 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.cn.1.4 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.cz.2.4 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.df.2.4 | $48$ | $2$ | $2$ | $1$ |
| 48.288.8-48.hy.2.3 | $48$ | $3$ | $3$ | $8$ |
| 48.384.7-48.gt.2.20 | $48$ | $4$ | $4$ | $7$ |
| 240.192.1-240.wp.2.7 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.wx.2.4 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.xv.2.4 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.yd.1.4 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.zb.2.4 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.zj.2.7 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bah.1.4 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bap.2.4 | $240$ | $2$ | $2$ | $1$ |
| 240.480.16-240.fs.2.28 | $240$ | $5$ | $5$ | $16$ |