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: | $\begin{bmatrix}23&48\\70&97\end{bmatrix}$, $\begin{bmatrix}25&8\\107&9\end{bmatrix}$, $\begin{bmatrix}31&40\\86&7\end{bmatrix}$, $\begin{bmatrix}41&8\\7&83\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 120.48.0.eo.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 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 |
|---|---|---|---|---|---|
| 24.48.0-8.bb.2.3 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-8.bb.2.4 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 120.48.0-120.dh.1.15 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-120.dh.1.16 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-120.ei.1.7 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-120.ei.1.11 | $120$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 120.288.8-120.tn.2.8 | $120$ | $3$ | $3$ | $8$ |
| 120.384.7-120.my.2.6 | $120$ | $4$ | $4$ | $7$ |
| 120.480.16-120.gj.2.3 | $120$ | $5$ | $5$ | $16$ |
| 240.192.1-240.kc.2.10 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.ki.1.12 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.ks.1.10 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.ky.2.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.uk.2.2 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.um.1.6 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.us.1.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.uu.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bao.1.14 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.baq.2.6 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.baw.2.12 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bay.1.16 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bdw.1.13 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bec.2.3 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bem.2.6 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bes.1.8 | $240$ | $2$ | $2$ | $1$ |