Invariants
| Level: | $12$ | $\SL_2$-level: | $4$ | ||||
| Index: | $4$ | $\PSL_2$-index: | $2$ | ||||
| Genus: | $0 = 1 + \frac{ 2 }{12} - \frac{ 0 }{4} - \frac{ 2 }{3} - \frac{ 1 }{2}$ | ||||||
| Cusps: | $1$ (which is rational) | Cusp widths | $2$ | Cusp orbits | $1$ | ||
| Elliptic points: | $0$ of order $2$ and $2$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $1$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 2A0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.4.0.1 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}3&2\\2&11\end{bmatrix}$, $\begin{bmatrix}5&1\\5&0\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 2.2.0.a.1 for the level structure with $-I$) |
| Cyclic 12-isogeny field degree: | $24$ |
| Cyclic 12-torsion field degree: | $96$ |
| Full 12-torsion field degree: | $1152$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 32740 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 2 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{2}(x^{2}+1728y^{2})}{y^{2}x^{2}}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 12.12.0-2.a.1.2 | $12$ | $3$ | $3$ | $0$ |
| 12.12.1-6.a.1.2 | $12$ | $3$ | $3$ | $1$ |
| 12.16.0-4.a.1.1 | $12$ | $4$ | $4$ | $0$ |
| 12.16.0-6.a.1.2 | $12$ | $4$ | $4$ | $0$ |
| 36.12.0-18.a.1.1 | $36$ | $3$ | $3$ | $0$ |
| 36.12.0-18.a.1.2 | $36$ | $3$ | $3$ | $0$ |
| 36.108.2-18.a.1.1 | $36$ | $27$ | $27$ | $2$ |
| 60.20.0-10.a.1.1 | $60$ | $5$ | $5$ | $0$ |
| 60.24.1-10.a.1.4 | $60$ | $6$ | $6$ | $1$ |
| 60.40.1-10.a.1.4 | $60$ | $10$ | $10$ | $1$ |
| 84.12.0-14.a.1.3 | $84$ | $3$ | $3$ | $0$ |
| 84.12.0-14.a.1.4 | $84$ | $3$ | $3$ | $0$ |
| 84.32.0-14.a.1.4 | $84$ | $8$ | $8$ | $0$ |
| 84.84.3-14.a.1.2 | $84$ | $21$ | $21$ | $3$ |
| 84.112.3-14.a.1.3 | $84$ | $28$ | $28$ | $3$ |
| 132.48.2-22.a.1.4 | $132$ | $12$ | $12$ | $2$ |
| 132.220.5-22.a.1.2 | $132$ | $55$ | $55$ | $5$ |
| 132.220.7-22.a.1.2 | $132$ | $55$ | $55$ | $7$ |
| 132.264.9-22.a.1.1 | $132$ | $66$ | $66$ | $9$ |
| 156.12.0-26.a.1.3 | $156$ | $3$ | $3$ | $0$ |
| 156.12.0-26.a.1.4 | $156$ | $3$ | $3$ | $0$ |
| 156.56.1-26.a.1.1 | $156$ | $14$ | $14$ | $1$ |
| 156.312.11-26.a.1.4 | $156$ | $78$ | $78$ | $11$ |
| 156.364.10-26.a.1.2 | $156$ | $91$ | $91$ | $10$ |
| 156.364.12-26.a.1.4 | $156$ | $91$ | $91$ | $12$ |
| 204.72.3-34.a.1.4 | $204$ | $18$ | $18$ | $3$ |
| 204.544.19-34.a.1.3 | $204$ | $136$ | $136$ | $19$ |
| 228.12.0-38.a.1.1 | $228$ | $3$ | $3$ | $0$ |
| 228.12.0-38.a.1.4 | $228$ | $3$ | $3$ | $0$ |
| 228.80.2-38.a.1.3 | $228$ | $20$ | $20$ | $2$ |
| 252.12.0-126.a.1.1 | $252$ | $3$ | $3$ | $0$ |
| 252.12.0-126.a.1.8 | $252$ | $3$ | $3$ | $0$ |
| 252.12.0-126.b.1.3 | $252$ | $3$ | $3$ | $0$ |
| 252.12.0-126.b.1.6 | $252$ | $3$ | $3$ | $0$ |
| 276.96.4-46.a.1.4 | $276$ | $24$ | $24$ | $4$ |