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: | $\begin{bmatrix}45&16\\16&65\end{bmatrix}$, $\begin{bmatrix}59&88\\50&71\end{bmatrix}$, $\begin{bmatrix}67&72\\30&49\end{bmatrix}$, $\begin{bmatrix}97&96\\4&113\end{bmatrix}$, $\begin{bmatrix}115&108\\42&11\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 120.48.0.bw.2 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $48$ |
| 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 but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.48.0-8.e.2.13 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 120.48.0-8.e.2.10 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-120.u.1.36 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-120.u.1.37 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-120.x.1.25 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-120.x.1.28 | $120$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 120.192.1-120.bz.1.2 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.ce.1.3 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.cy.2.9 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.db.2.2 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.ea.2.10 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.eb.2.1 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.eg.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.eh.1.4 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.gq.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.gr.1.7 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.gs.2.11 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.gt.2.1 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.hg.2.9 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.hh.2.3 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.hi.1.3 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.hj.1.5 | $120$ | $2$ | $2$ | $1$ |
| 120.288.8-120.mj.1.52 | $120$ | $3$ | $3$ | $8$ |
| 120.384.7-120.gv.1.57 | $120$ | $4$ | $4$ | $7$ |
| 120.480.16-120.cl.1.30 | $120$ | $5$ | $5$ | $16$ |