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: | $\begin{bmatrix}30&29\\109&110\end{bmatrix}$, $\begin{bmatrix}54&31\\1&72\end{bmatrix}$, $\begin{bmatrix}71&103\\39&58\end{bmatrix}$, $\begin{bmatrix}115&24\\19&59\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 24.8.0.d.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 $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -\frac{3^3}{2^9}\cdot\frac{x^{8}(x^{2}-72y^{2})(x^{2}-8y^{2})^{3}}{y^{6}x^{10}}$ |
Modular covers
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.1.6 | $120$ | $2$ | $2$ | $0$ | $?$ |
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.cb.1.1 | $120$ | $3$ | $3$ | $0$ |
120.48.1-24.ci.1.1 | $120$ | $3$ | $3$ | $1$ |
120.64.1-24.d.1.5 | $120$ | $4$ | $4$ | $1$ |
120.80.2-120.d.1.13 | $120$ | $5$ | $5$ | $2$ |
120.96.3-120.bt.1.31 | $120$ | $6$ | $6$ | $3$ |
120.160.5-120.d.1.32 | $120$ | $10$ | $10$ | $5$ |