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 | $1^{2}\cdot2\cdot3^{2}\cdot4^{2}\cdot6\cdot12^{2}$ | 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 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.96.0.23 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}1&2\\0&7\end{bmatrix}$, $\begin{bmatrix}1&8\\0&5\end{bmatrix}$, $\begin{bmatrix}1&11\\0&1\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $S_3\times D_4$ |
| Contains $-I$: | no $\quad$ (see 12.48.0.c.1 for the level structure with $-I$) |
| Cyclic 12-isogeny field degree: | $1$ |
| Cyclic 12-torsion field degree: | $1$ |
| 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{x^{48}(x^{4}+4x^{3}y-8xy^{3}-8y^{4})^{3}(x^{12}+12x^{11}y+72x^{10}y^{2}+280x^{9}y^{3}+792x^{8}y^{4}+1728x^{7}y^{5}+2880x^{6}y^{6}+3456x^{5}y^{7}+2496x^{4}y^{8}+256x^{3}y^{9}-1536x^{2}y^{10}-1536xy^{11}-512y^{12})^{3}}{y^{6}x^{60}(x+y)^{2}(x+2y)^{12}(x^{2}+2xy-2y^{2})(x^{2}+2xy+2y^{2})^{3}(x^{2}+2xy+4y^{2})^{4}}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_1(3)$ | $3$ | $12$ | $12$ | $0$ | $0$ |
| $X_1(4)$ | $4$ | $8$ | $8$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.48.0-12.g.1.1 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 12.48.0-12.g.1.12 | $12$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.