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: | 48.96.0.968 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}15&29\\40&25\end{bmatrix}$, $\begin{bmatrix}17&41\\44&25\end{bmatrix}$, $\begin{bmatrix}19&35\\12&35\end{bmatrix}$, $\begin{bmatrix}39&44\\16&3\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.0.bq.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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^{11}}{3^2}\cdot\frac{(6x+y)^{48}(9889579008x^{16}-6879707136x^{15}y-3153199104x^{14}y^{2}+13759414272x^{13}y^{3}-6628884480x^{12}y^{4}-5303107584x^{11}y^{5}+17149501440x^{10}y^{6}-20312653824x^{9}y^{7}+17039227392x^{8}y^{8}-10329845760x^{7}y^{9}+4787679744x^{6}y^{10}-1745473536x^{5}y^{11}+508057920x^{4}y^{12}-117561600x^{3}y^{13}+20876640x^{2}y^{14}-2896512xy^{15}+264263y^{16})^{3}}{(6x+y)^{48}(12x^{2}-12xy-y^{2})^{16}(12x^{2}-4xy+3y^{2})^{4}(12x^{2}+12xy-5y^{2})^{2}(720x^{4}-864x^{3}y+504x^{2}y^{2}-24xy^{3}+29y^{4})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 16.48.0-16.f.2.1 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-24.by.2.13 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-16.f.2.12 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.h.1.19 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.h.1.24 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-24.by.2.7 | $48$ | $2$ | $2$ | $0$ | $0$ |
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.m.1.2 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.bb.2.1 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.bp.2.3 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.bz.2.2 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.ch.1.1 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.cu.2.1 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.cy.2.3 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.dj.2.2 | $48$ | $2$ | $2$ | $1$ |
| 48.288.8-48.iq.1.5 | $48$ | $3$ | $3$ | $8$ |
| 48.384.7-48.hd.2.6 | $48$ | $4$ | $4$ | $7$ |
| 240.192.1-240.xh.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.xp.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.yn.2.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.yv.2.2 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.zt.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bab.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.baz.2.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bbh.2.2 | $240$ | $2$ | $2$ | $1$ |
| 240.480.16-240.gk.1.4 | $240$ | $5$ | $5$ | $16$ |