Invariants
| Level: | $120$ | $\SL_2$-level: | $8$ | ||||
| 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 | $2^{4}\cdot4^{2}\cdot8^{4}$ | 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: | 8O0 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}1&32\\59&21\end{bmatrix}$, $\begin{bmatrix}7&64\\78&77\end{bmatrix}$, $\begin{bmatrix}23&56\\20&57\end{bmatrix}$, $\begin{bmatrix}87&64\\31&103\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.48.0.bn.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $24$ |
| Cyclic 120-torsion field degree: | $768$ |
| Full 120-torsion field degree: | $368640$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 7 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3}\cdot\frac{(3x-y)^{48}(5186654208x^{16}-24078974976x^{15}y+51257401344x^{14}y^{2}-66485919744x^{13}y^{3}+58862702592x^{12}y^{4}-37721935872x^{11}y^{5}+18090335232x^{10}y^{6}-6614203392x^{9}y^{7}+1859283072x^{8}y^{8}-401780736x^{7}y^{9}+66151296x^{6}y^{10}-8138880x^{5}y^{11}+724032x^{4}y^{12}-44352x^{3}y^{13}+1776x^{2}y^{14}-48xy^{15}+y^{16})^{3}}{x^{2}(3x-y)^{50}(6x^{2}-6xy+y^{2})^{2}(12x^{2}-6xy+y^{2})^{4}(72x^{4}-36x^{2}y^{2}+12xy^{3}-y^{4})^{8}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.48.0-8.bb.2.7 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 120.48.0-8.bb.2.1 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-24.bl.1.3 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-24.bl.1.4 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-24.by.1.2 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-24.by.1.14 | $120$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 120.288.8-24.gr.2.1 | $120$ | $3$ | $3$ | $8$ |
| 120.384.7-24.eq.2.11 | $120$ | $4$ | $4$ | $7$ |
| 120.480.16-120.ez.2.2 | $120$ | $5$ | $5$ | $16$ |
| 240.192.1-48.cw.2.8 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-48.cy.1.2 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-48.de.1.3 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-48.dg.2.8 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-48.ec.1.3 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-48.ee.2.7 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-48.ek.2.7 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-48.em.1.2 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.ms.2.15 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.mu.1.11 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.na.1.12 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.nc.2.16 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.rq.1.3 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.rs.2.11 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.ry.2.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.sa.1.1 | $240$ | $2$ | $2$ | $1$ |