Invariants
| Level: | $120$ | $\SL_2$-level: | $6$ | ||||
| Index: | $16$ | $\PSL_2$-index: | $8$ | ||||
| Genus: | $0 = 1 + \frac{ 8 }{12} - \frac{ 0 }{4} - \frac{ 2 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $2\cdot6$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $2$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 6C0 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}4&111\\27&4\end{bmatrix}$, $\begin{bmatrix}5&34\\9&73\end{bmatrix}$, $\begin{bmatrix}9&107\\82&107\end{bmatrix}$, $\begin{bmatrix}36&37\\79&105\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.8.0.b.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $72$ |
| Cyclic 120-torsion field degree: | $2304$ |
| Full 120-torsion field degree: | $2211840$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 223 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 8 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^3}\cdot\frac{(x+y)^{8}(x^{2}+6y^{2})^{3}(x^{2}+54y^{2})}{y^{6}x^{2}(x+y)^{8}}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.2.0.b.1 | $8$ | $8$ | $4$ | $0$ | $0$ |
| 15.8.0-3.a.1.2 | $15$ | $2$ | $2$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 15.8.0-3.a.1.2 | $15$ | $2$ | $2$ | $0$ | $0$ |
| 120.8.0-3.a.1.5 | $120$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 120.48.0-24.y.1.1 | $120$ | $3$ | $3$ | $0$ |
| 120.48.1-24.cd.1.1 | $120$ | $3$ | $3$ | $1$ |
| 120.64.1-24.b.1.5 | $120$ | $4$ | $4$ | $1$ |
| 120.80.2-120.b.1.13 | $120$ | $5$ | $5$ | $2$ |
| 120.96.3-120.d.1.30 | $120$ | $6$ | $6$ | $3$ |
| 120.160.5-120.b.1.32 | $120$ | $10$ | $10$ | $5$ |