Invariants
| Level: | $120$ | $\SL_2$-level: | $40$ | Newform level: | $1$ | ||
| Index: | $240$ | $\PSL_2$-index: | $120$ | ||||
| Genus: | $8 = 1 + \frac{ 120 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $10^{4}\cdot40^{2}$ | Cusp orbits | $2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 14$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 8$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 40A8 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}17&39\\40&61\end{bmatrix}$, $\begin{bmatrix}37&61\\60&43\end{bmatrix}$, $\begin{bmatrix}55&14\\4&103\end{bmatrix}$, $\begin{bmatrix}67&13\\116&51\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 120.120.8.cy.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $48$ |
| Cyclic 120-torsion field degree: | $768$ |
| Full 120-torsion field degree: | $147456$ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_{S_4}(5)$ | $5$ | $48$ | $24$ | $0$ | $0$ |
| 24.48.0-24.be.1.8 | $24$ | $5$ | $5$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.48.0-24.be.1.8 | $24$ | $5$ | $5$ | $0$ | $0$ |
| 40.120.4-40.w.1.8 | $40$ | $2$ | $2$ | $4$ | $2$ |
| 120.120.4-40.w.1.4 | $120$ | $2$ | $2$ | $4$ | $?$ |
| 120.120.4-120.cc.1.4 | $120$ | $2$ | $2$ | $4$ | $?$ |
| 120.120.4-120.cc.1.23 | $120$ | $2$ | $2$ | $4$ | $?$ |
| 120.120.4-120.cd.1.12 | $120$ | $2$ | $2$ | $4$ | $?$ |
| 120.120.4-120.cd.1.31 | $120$ | $2$ | $2$ | $4$ | $?$ |