Invariants
Level: | $120$ | $\SL_2$-level: | $8$ | ||||
Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $2^{4}\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
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: | 8G0 |
Level structure
$\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}15&56\\23&85\end{bmatrix}$, $\begin{bmatrix}23&8\\111&25\end{bmatrix}$, $\begin{bmatrix}25&16\\16&59\end{bmatrix}$, $\begin{bmatrix}77&8\\114&73\end{bmatrix}$, $\begin{bmatrix}91&8\\26&77\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 40.24.0.bj.1 for the level structure with $-I$) |
Cyclic 120-isogeny field degree: | $24$ |
Cyclic 120-torsion field degree: | $768$ |
Full 120-torsion field degree: | $737280$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 44 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^4\cdot5}\cdot\frac{x^{24}(625x^{8}+480000x^{6}y^{2}+13721600x^{4}y^{4}+78643200x^{2}y^{6}+16777216y^{8})^{3}}{y^{2}x^{26}(5x^{2}-64y^{2})^{8}(5x^{2}+64y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
24.24.0-8.n.1.7 | $24$ | $2$ | $2$ | $0$ | $0$ |
120.24.0-8.n.1.4 | $120$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.