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 | $2^{2}\cdot4^{2}$ | 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: | 4E0 |
Level structure
$\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}1&76\\21&115\end{bmatrix}$, $\begin{bmatrix}35&116\\116&67\end{bmatrix}$, $\begin{bmatrix}39&56\\38&101\end{bmatrix}$, $\begin{bmatrix}55&24\\112&89\end{bmatrix}$, $\begin{bmatrix}107&92\\26&93\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 12.12.0.h.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 771 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{3}\cdot\frac{(3x-2y)^{12}(9x^{4}+42x^{2}y^{2}+y^{4})^{3}}{y^{2}x^{2}(3x-2y)^{12}(3x^{2}-y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
40.12.0-4.c.1.6 | $40$ | $2$ | $2$ | $0$ | $0$ |
120.12.0-4.c.1.4 | $120$ | $2$ | $2$ | $0$ | $?$ |
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.bk.1.5 | $120$ | $2$ | $2$ | $0$ |
120.48.0-24.bk.1.8 | $120$ | $2$ | $2$ | $0$ |
120.48.0-24.bl.1.10 | $120$ | $2$ | $2$ | $0$ |
120.48.0-24.bl.1.12 | $120$ | $2$ | $2$ | $0$ |
120.48.0-24.bs.1.6 | $120$ | $2$ | $2$ | $0$ |
120.48.0-24.bs.1.7 | $120$ | $2$ | $2$ | $0$ |
120.48.0-24.bt.1.6 | $120$ | $2$ | $2$ | $0$ |
120.48.0-24.bt.1.8 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.cc.1.9 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.cc.1.15 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.cd.1.11 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.cd.1.13 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.cg.1.11 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.cg.1.13 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.ch.1.9 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.ch.1.15 | $120$ | $2$ | $2$ | $0$ |
120.72.2-12.t.1.8 | $120$ | $3$ | $3$ | $2$ |
120.96.1-12.l.1.9 | $120$ | $4$ | $4$ | $1$ |
120.120.4-60.l.1.8 | $120$ | $5$ | $5$ | $4$ |
120.144.3-60.et.1.9 | $120$ | $6$ | $6$ | $3$ |
120.240.7-60.t.1.15 | $120$ | $10$ | $10$ | $7$ |