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 | $2^{8}\cdot16^{2}$ | 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: | 16G0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.96.0.1459 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}25&1\\12&1\end{bmatrix}$, $\begin{bmatrix}39&47\\20&43\end{bmatrix}$, $\begin{bmatrix}43&20\\44&45\end{bmatrix}$, $\begin{bmatrix}45&19\\40&47\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.0.w.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 4 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2\cdot3}\cdot\frac{(6x+5y)^{48}(674319602647296x^{16}+114449813581824x^{15}y-132029317370880x^{14}y^{2}-801594877224960x^{13}y^{3}-2781598349134080x^{12}y^{4}-5988850748310528x^{11}y^{5}-6832811827360512x^{10}y^{6}+1830309943795200x^{9}y^{7}+26878088483816544x^{8}y^{8}+74006021508104448x^{7}y^{9}+144924629893081920x^{6}y^{10}+220409689687060608x^{5}y^{11}+256046921933589360x^{4}y^{12}+219652351431218496x^{3}y^{13}+131228670383028912x^{2}y^{14}+48782844227390112xy^{15}+8500545331353601y^{16})^{3}}{(x-2y)^{2}(6x+5y)^{50}(30x^{2}+84xy+y^{2})^{16}(42x^{2}+36xy+49y^{2})^{2}(2628x^{4}+1008x^{3}y+3708x^{2}y^{2}+6888xy^{3}+4801y^{4})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 16.48.0-16.e.2.3 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-24.bj.1.9 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-16.e.2.7 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.e.1.4 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.e.1.17 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-24.bj.1.6 | $48$ | $2$ | $2$ | $0$ | $0$ |
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.cj.1.3 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.ck.2.3 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.cr.1.5 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.cs.2.6 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.dp.1.2 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.dq.1.4 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.dx.1.2 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.dy.1.3 | $48$ | $2$ | $2$ | $1$ |
| 48.288.8-48.dg.2.1 | $48$ | $3$ | $3$ | $8$ |
| 48.384.7-48.ee.1.25 | $48$ | $4$ | $4$ | $7$ |
| 240.192.1-240.lz.1.3 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.ma.1.3 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.mh.1.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.mi.1.3 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.qx.1.3 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.qy.1.3 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.rf.1.3 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.rg.1.5 | $240$ | $2$ | $2$ | $1$ |
| 240.480.16-240.by.2.10 | $240$ | $5$ | $5$ | $16$ |