Invariants
Level: | $120$ | $\SL_2$-level: | $8$ | ||||
Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $1^{2}\cdot2\cdot8$ | Cusp orbits | $1^{2}\cdot2$ | ||
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: | 8C0 |
Level structure
$\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}1&114\\80&19\end{bmatrix}$, $\begin{bmatrix}27&62\\4&33\end{bmatrix}$, $\begin{bmatrix}34&43\\15&118\end{bmatrix}$, $\begin{bmatrix}57&26\\14&85\end{bmatrix}$, $\begin{bmatrix}72&5\\17&68\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 40.12.0.y.1 for the level structure with $-I$) |
Cyclic 120-isogeny field degree: | $48$ |
Cyclic 120-torsion field degree: | $1536$ |
Full 120-torsion field degree: | $1474560$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 634 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^{20}\cdot5}\cdot\frac{x^{12}(25x^{4}+2560x^{2}y^{2}+16384y^{4})^{3}}{y^{8}x^{14}(5x^{2}+512y^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
24.12.0-4.c.1.6 | $24$ | $2$ | $2$ | $0$ | $0$ |
60.12.0-4.c.1.1 | $60$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
120.48.0-40.l.1.12 | $120$ | $2$ | $2$ | $0$ |
120.48.0-40.n.1.5 | $120$ | $2$ | $2$ | $0$ |
120.48.0-40.v.1.4 | $120$ | $2$ | $2$ | $0$ |
120.48.0-40.x.1.3 | $120$ | $2$ | $2$ | $0$ |
120.48.0-40.bm.1.5 | $120$ | $2$ | $2$ | $0$ |
120.48.0-40.bp.1.5 | $120$ | $2$ | $2$ | $0$ |
120.48.0-40.bt.1.1 | $120$ | $2$ | $2$ | $0$ |
120.48.0-40.bu.1.3 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.bu.1.6 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.bw.1.10 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.cc.1.10 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.ce.1.10 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.dk.1.9 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.dn.1.9 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.dt.1.2 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.du.1.9 | $120$ | $2$ | $2$ | $0$ |
120.72.2-120.cu.1.63 | $120$ | $3$ | $3$ | $2$ |
120.96.1-120.zu.1.37 | $120$ | $4$ | $4$ | $1$ |
120.120.4-40.bm.1.13 | $120$ | $5$ | $5$ | $4$ |
120.144.3-40.by.1.11 | $120$ | $6$ | $6$ | $3$ |
120.240.7-40.ck.1.17 | $120$ | $10$ | $10$ | $7$ |