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: | $10$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | 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: | 8O0 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}5&56\\102&109\end{bmatrix}$, $\begin{bmatrix}67&48\\42&35\end{bmatrix}$, $\begin{bmatrix}69&68\\56&39\end{bmatrix}$, $\begin{bmatrix}103&68\\14&3\end{bmatrix}$, $\begin{bmatrix}109&32\\76&75\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.48.0.x.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $48$ |
| Cyclic 120-torsion field degree: | $1536$ |
| 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 3 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{3^2\cdot5^8}\cdot\frac{(4x+y)^{48}(1675264x^{8}+3575808x^{7}y-1947648x^{6}y^{2}-2916864x^{5}y^{3}+7217280x^{4}y^{4}-3115584x^{3}y^{5}+927072x^{2}y^{6}-414072xy^{7}+76869y^{8})^{3}(3887104x^{8}-7852032x^{7}y+6592512x^{6}y^{2}-8308224x^{5}y^{3}+7217280x^{4}y^{4}-1093824x^{3}y^{5}-273888x^{2}y^{6}+188568xy^{7}+33129y^{8})^{3}}{(4x+y)^{48}(8x^{2}-3y^{2})^{8}(8x^{2}-36xy+3y^{2})^{8}(24x^{2}-8xy+9y^{2})^{4}(1472x^{4}+1152x^{3}y-3888x^{2}y^{2}+432xy^{3}+207y^{4})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.48.0-8.h.1.7 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 120.48.0-8.h.1.3 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-24.h.1.7 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-24.h.1.19 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-24.h.2.1 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-24.h.2.13 | $120$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 120.192.1-24.e.1.4 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-24.e.2.1 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-24.k.1.7 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-24.k.2.5 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-24.bu.1.3 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-24.bu.2.1 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-24.bv.1.7 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-24.bv.2.1 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.ok.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.ok.2.16 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.ol.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.ol.2.14 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.pc.1.7 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.pc.2.6 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.pd.1.5 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.pd.2.6 | $120$ | $2$ | $2$ | $1$ |
| 120.288.8-24.et.1.3 | $120$ | $3$ | $3$ | $8$ |
| 120.384.7-24.dd.1.11 | $120$ | $4$ | $4$ | $7$ |
| 120.480.16-120.ds.1.18 | $120$ | $5$ | $5$ | $16$ |