Invariants
Level: | $120$ | $\SL_2$-level: | $24$ | ||||
Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
Cusps: | $10$ (of which $4$ are rational) | Cusp widths | $1^{2}\cdot2\cdot3^{2}\cdot4^{2}\cdot6\cdot12^{2}$ | Cusp orbits | $1^{4}\cdot2^{3}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $1$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $4$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 12J0 |
Level structure
$\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}12&89\\89&60\end{bmatrix}$, $\begin{bmatrix}45&58\\82&33\end{bmatrix}$, $\begin{bmatrix}61&72\\52&65\end{bmatrix}$, $\begin{bmatrix}75&76\\4&15\end{bmatrix}$, $\begin{bmatrix}86&7\\83&54\end{bmatrix}$, $\begin{bmatrix}113&22\\80&27\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 12.48.0.c.3 for the level structure with $-I$) |
Cyclic 120-isogeny field degree: | $12$ |
Cyclic 120-torsion field degree: | $384$ |
Full 120-torsion field degree: | $368640$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 17 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^3}\cdot\frac{(x-2y)^{48}(x^{4}+8x^{3}y-24x^{2}y^{2}+32xy^{3}+16y^{4})^{3}(x^{12}-24x^{11}y+312x^{10}y^{2}-1504x^{9}y^{3}+1776x^{8}y^{4}+8448x^{7}y^{5}-28416x^{6}y^{6}+33792x^{5}y^{7}+28416x^{4}y^{8}-96256x^{3}y^{9}+79872x^{2}y^{10}-24576xy^{11}+4096y^{12})^{3}}{y^{3}x^{3}(x-2y)^{54}(x+2y)^{2}(x^{2}+4y^{2})^{12}(x^{2}-8xy+4y^{2})^{4}(x^{2}-2xy+4y^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
120.48.0-12.g.1.13 | $120$ | $2$ | $2$ | $0$ | $?$ |
120.48.0-12.g.1.21 | $120$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.