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: | 48.96.0.470 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}7&31\\28&15\end{bmatrix}$, $\begin{bmatrix}11&35\\12&31\end{bmatrix}$, $\begin{bmatrix}11&37\\12&7\end{bmatrix}$, $\begin{bmatrix}33&46\\4&39\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.0.z.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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^2}{3\cdot5^2}\cdot\frac{(2x-y)^{48}(1869005056x^{16}+5079877632x^{15}y+6471444480x^{14}y^{2}+5119994880x^{13}y^{3}+2709745920x^{12}y^{4}+229049856x^{11}y^{5}-4174668288x^{10}y^{6}-13753255680x^{9}y^{7}-24041227680x^{8}y^{8}-20629883520x^{7}y^{9}-9393003648x^{6}y^{10}+773043264x^{5}y^{11}+13718088720x^{4}y^{12}+38879961120x^{3}y^{13}+73713797280x^{2}y^{14}+86794471728xy^{15}+47900555361y^{16})^{3}}{(2x-y)^{48}(2x^{2}-3y^{2})^{2}(2x^{2}+18xy+3y^{2})^{16}(6x^{2}+4xy+9y^{2})^{2}(52x^{4}+36x^{3}y+18x^{2}y^{2}+54xy^{3}+117y^{4})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 16.48.0-16.f.1.4 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-24.bl.1.5 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-16.f.1.6 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.f.2.10 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.f.2.15 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-24.bl.1.5 | $48$ | $2$ | $2$ | $0$ | $0$ |
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.cx.1.8 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.cy.1.8 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.df.1.6 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.dg.1.7 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.ed.1.3 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.ee.1.5 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.el.1.6 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.em.1.4 | $48$ | $2$ | $2$ | $1$ |
| 48.288.8-48.dn.2.8 | $48$ | $3$ | $3$ | $8$ |
| 48.384.7-48.ek.2.9 | $48$ | $4$ | $4$ | $7$ |
| 240.192.1-240.mv.1.8 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.mw.1.12 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.nd.1.10 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.ne.1.7 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.rt.1.7 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.ru.1.10 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.sb.1.12 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.sc.1.8 | $240$ | $2$ | $2$ | $1$ |
| 240.480.16-240.cf.2.15 | $240$ | $5$ | $5$ | $16$ |