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 $2$ are rational) | Cusp widths | $2^{4}\cdot4\cdot6^{4}\cdot12$ | Cusp orbits | $1^{2}\cdot2^{2}\cdot4$ | ||
| 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: | 12.96.0.39 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}1&3\\0&11\end{bmatrix}$, $\begin{bmatrix}1&3\\6&1\end{bmatrix}$, $\begin{bmatrix}1&7\\0&7\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $S_3\times D_4$ |
| Contains $-I$: | no $\quad$ (see 12.48.0.b.1 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 9 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^2}\cdot\frac{x^{48}(x^{4}+12x^{3}y-18x^{2}y^{2}+12xy^{3}-3y^{4})^{3}(601x^{12}-3324x^{11}y+9642x^{10}y^{2}-17100x^{9}y^{3}+19755x^{8}y^{4}-14424x^{7}y^{5}+5628x^{6}y^{6}-24x^{5}y^{7}-1185x^{4}y^{8}+660x^{3}y^{9}-198x^{2}y^{10}+36xy^{11}-3y^{12})^{3}}{x^{52}(x-y)^{12}(x^{2}+y^{2})^{6}(5x^{2}-4xy+y^{2})^{6}(13x^{4}-36x^{3}y+54x^{2}y^{2}-36xy^{3}+9y^{4})^{2}}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_1(3)$ | $3$ | $12$ | $12$ | $0$ | $0$ |
| 4.12.0-4.a.1.1 | $4$ | $8$ | $8$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.48.0-12.d.1.1 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 12.48.0-12.d.1.11 | $12$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.