Invariants
| Level: | $12$ | $\SL_2$-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 $4$ are rational) | Cusp widths | $2^{4}\cdot4\cdot6^{4}\cdot12$ | 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: | 12I0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.96.0.52 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}5&2\\6&11\end{bmatrix}$, $\begin{bmatrix}5&10\\0&1\end{bmatrix}$, $\begin{bmatrix}7&0\\0&11\end{bmatrix}$, $\begin{bmatrix}7&0\\6&11\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $C_2^2\times D_6$ |
| Contains $-I$: | no $\quad$ (see 12.48.0.a.2 for the level structure with $-I$) |
| Cyclic 12-isogeny field degree: | $2$ |
| Cyclic 12-torsion field degree: | $4$ |
| Full 12-torsion field degree: | $48$ |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{(x+2y)^{48}(x^{4}+2x^{3}y+6x^{2}y^{2}+8xy^{3}+4y^{4})^{3}(x^{12}+6x^{11}y+246x^{10}y^{2}+1400x^{9}y^{3}+3960x^{8}y^{4}+6696x^{7}y^{5}+7224x^{6}y^{6}+5184x^{5}y^{7}+2880x^{4}y^{8}+1760x^{3}y^{9}+1056x^{2}y^{10}+384xy^{11}+64y^{12})^{3}}{y^{2}x^{4}(x+y)^{2}(x+2y)^{60}(x^{2}-2xy-2y^{2})^{6}(x^{2}+xy+y^{2})^{6}(x^{2}+2xy+2y^{2})^{2}}$ |
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 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 12.48.0-6.a.1.6 | $12$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.