Invariants
| Level: | $60$ | $\SL_2$-level: | $12$ | ||||
| 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 $2$ are rational) | Cusp widths | $2^{4}\cdot4\cdot6^{4}\cdot12$ | Cusp orbits | $1^{2}\cdot2^{4}$ | ||
| 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: | 12I0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.96.0.100 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}17&54\\42&41\end{bmatrix}$, $\begin{bmatrix}19&30\\36&13\end{bmatrix}$, $\begin{bmatrix}37&20\\24&7\end{bmatrix}$, $\begin{bmatrix}37&50\\0&37\end{bmatrix}$, $\begin{bmatrix}41&44\\30&43\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.48.0.a.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $12$ |
| Cyclic 60-torsion field degree: | $96$ |
| Full 60-torsion field degree: | $23040$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 12 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^{12}\cdot3^6\cdot5^2}\cdot\frac{(2x+3y)^{48}(208x^{4}+448x^{3}y+1608x^{2}y^{2}+1408xy^{3}+553y^{4})^{3}(5459968x^{12}+82919424x^{11}y+449845248x^{10}y^{2}+888586240x^{9}y^{3}+970018560x^{8}y^{4}+495212544x^{7}y^{5}+661661952x^{6}y^{6}+1595188224x^{5}y^{7}+2146278960x^{4}y^{8}+1406712640x^{3}y^{9}+428201688x^{2}y^{10}+105181824xy^{11}+33623353y^{12})^{3}}{(2x-y)^{4}(2x+3y)^{60}(2x^{2}+xy+2y^{2})^{6}(4x^{2}-28xy-11y^{2})^{6}(4x^{2}+32xy+19y^{2})^{2}(28x^{2}+44xy+43y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.48.0-6.a.1.3 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 60.48.0-6.a.1.6 | $60$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.