Invariants
| Level: | $12$ | $\SL_2$-level: | $12$ | ||||
| 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 | $1^{2}\cdot3^{2}\cdot4\cdot12$ | 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: | yes $\quad(D =$ $-3$) |
Other labels
| Cummins and Pauli (CP) label: | 12E0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.48.0.93 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}11&0\\6&7\end{bmatrix}$, $\begin{bmatrix}11&2\\6&5\end{bmatrix}$, $\begin{bmatrix}11&3\\0&11\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $C_2^2:D_{12}$ |
| Contains $-I$: | no $\quad$ (see 12.24.0.i.1 for the level structure with $-I$) |
| Cyclic 12-isogeny field degree: | $2$ |
| Cyclic 12-torsion field degree: | $8$ |
| Full 12-torsion field degree: | $96$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 85 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^4\cdot3^6}\cdot\frac{(3x-2y)^{3}(3x+y)^{24}(3x+2y)^{3}(9x^{3}-18x^{2}y+12xy^{2}+8y^{3})^{3}(9x^{3}+18x^{2}y+12xy^{2}-8y^{3})^{3}}{y^{4}x^{12}(3x+y)^{24}(3x^{2}-4y^{2})^{3}(27x^{2}-4y^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.24.0-6.a.1.1 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 12.24.0-6.a.1.4 | $12$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.