Invariants
| Level: | $12$ | $\SL_2$-level: | $4$ | ||||
| Index: | $12$ | $\PSL_2$-index: | $6$ | ||||
| Genus: | $0 = 1 + \frac{ 6 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}$ | ||||||
| Cusps: | $3$ (all of which are rational) | Cusp widths | $2^{3}$ | Cusp orbits | $1^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $3$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 2C0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.12.0.2 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}1&10\\0&5\end{bmatrix}$, $\begin{bmatrix}3&4\\10&9\end{bmatrix}$, $\begin{bmatrix}9&2\\10&9\end{bmatrix}$, $\begin{bmatrix}9&10\\10&11\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $C_2^3\times \GL(2,3)$ |
| Contains $-I$: | no $\quad$ (see 2.6.0.a.1 for the level structure with $-I$) |
| Cyclic 12-isogeny field degree: | $8$ |
| Cyclic 12-torsion field degree: | $32$ |
| Full 12-torsion field degree: | $384$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 31720 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 6 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{6}(x^{2}+192y^{2})^{3}}{y^{2}x^{6}(x-8y)^{2}(x+8y)^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.4.0-2.a.1.1 | $12$ | $3$ | $3$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 12.24.0-4.a.1.2 | $12$ | $2$ | $2$ | $0$ |
| 12.24.0-4.b.1.2 | $12$ | $2$ | $2$ | $0$ |
| 12.36.1-6.a.1.6 | $12$ | $3$ | $3$ | $1$ |
| 12.48.0-6.a.1.10 | $12$ | $4$ | $4$ | $0$ |
| 24.24.0-8.a.1.4 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0-8.b.1.4 | $24$ | $2$ | $2$ | $0$ |
| 60.60.2-10.a.1.3 | $60$ | $5$ | $5$ | $2$ |
| 60.72.1-10.a.1.7 | $60$ | $6$ | $6$ | $1$ |
| 60.120.3-10.a.1.8 | $60$ | $10$ | $10$ | $3$ |
| 12.24.0-12.a.1.3 | $12$ | $2$ | $2$ | $0$ |
| 12.24.0-12.b.1.3 | $12$ | $2$ | $2$ | $0$ |
| 84.96.2-14.a.1.10 | $84$ | $8$ | $8$ | $2$ |
| 84.252.7-14.a.1.8 | $84$ | $21$ | $21$ | $7$ |
| 84.336.9-14.a.1.6 | $84$ | $28$ | $28$ | $9$ |
| 36.324.10-18.a.1.4 | $36$ | $27$ | $27$ | $10$ |
| 60.24.0-20.a.1.4 | $60$ | $2$ | $2$ | $0$ |
| 60.24.0-20.b.1.4 | $60$ | $2$ | $2$ | $0$ |
| 132.144.4-22.a.1.10 | $132$ | $12$ | $12$ | $4$ |
| 24.24.0-24.a.1.8 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0-24.b.1.7 | $24$ | $2$ | $2$ | $0$ |
| 156.168.5-26.a.1.5 | $156$ | $14$ | $14$ | $5$ |
| 84.24.0-28.a.1.4 | $84$ | $2$ | $2$ | $0$ |
| 84.24.0-28.b.1.4 | $84$ | $2$ | $2$ | $0$ |
| 204.216.7-34.a.1.5 | $204$ | $18$ | $18$ | $7$ |
| 228.240.8-38.a.1.10 | $228$ | $20$ | $20$ | $8$ |
| 120.24.0-40.a.1.8 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-40.b.1.8 | $120$ | $2$ | $2$ | $0$ |
| 132.24.0-44.a.1.3 | $132$ | $2$ | $2$ | $0$ |
| 132.24.0-44.b.1.4 | $132$ | $2$ | $2$ | $0$ |
| 276.288.10-46.a.1.10 | $276$ | $24$ | $24$ | $10$ |
| 156.24.0-52.a.1.4 | $156$ | $2$ | $2$ | $0$ |
| 156.24.0-52.b.1.4 | $156$ | $2$ | $2$ | $0$ |
| 168.24.0-56.a.1.6 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-56.b.1.8 | $168$ | $2$ | $2$ | $0$ |
| 60.24.0-60.a.1.7 | $60$ | $2$ | $2$ | $0$ |
| 60.24.0-60.b.1.8 | $60$ | $2$ | $2$ | $0$ |
| 204.24.0-68.a.1.4 | $204$ | $2$ | $2$ | $0$ |
| 204.24.0-68.b.1.4 | $204$ | $2$ | $2$ | $0$ |
| 228.24.0-76.a.1.4 | $228$ | $2$ | $2$ | $0$ |
| 228.24.0-76.b.1.4 | $228$ | $2$ | $2$ | $0$ |
| 84.24.0-84.a.1.5 | $84$ | $2$ | $2$ | $0$ |
| 84.24.0-84.b.1.6 | $84$ | $2$ | $2$ | $0$ |
| 264.24.0-88.a.1.7 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0-88.b.1.8 | $264$ | $2$ | $2$ | $0$ |
| 276.24.0-92.a.1.3 | $276$ | $2$ | $2$ | $0$ |
| 276.24.0-92.b.1.4 | $276$ | $2$ | $2$ | $0$ |
| 312.24.0-104.a.1.7 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-104.b.1.8 | $312$ | $2$ | $2$ | $0$ |
| 120.24.0-120.a.1.15 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-120.b.1.15 | $120$ | $2$ | $2$ | $0$ |
| 132.24.0-132.a.1.6 | $132$ | $2$ | $2$ | $0$ |
| 132.24.0-132.b.1.6 | $132$ | $2$ | $2$ | $0$ |
| 156.24.0-156.a.1.5 | $156$ | $2$ | $2$ | $0$ |
| 156.24.0-156.b.1.5 | $156$ | $2$ | $2$ | $0$ |
| 168.24.0-168.a.1.10 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-168.b.1.12 | $168$ | $2$ | $2$ | $0$ |
| 204.24.0-204.a.1.7 | $204$ | $2$ | $2$ | $0$ |
| 204.24.0-204.b.1.5 | $204$ | $2$ | $2$ | $0$ |
| 228.24.0-228.a.1.6 | $228$ | $2$ | $2$ | $0$ |
| 228.24.0-228.b.1.5 | $228$ | $2$ | $2$ | $0$ |
| 264.24.0-264.a.1.12 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0-264.b.1.14 | $264$ | $2$ | $2$ | $0$ |
| 276.24.0-276.a.1.6 | $276$ | $2$ | $2$ | $0$ |
| 276.24.0-276.b.1.5 | $276$ | $2$ | $2$ | $0$ |
| 312.24.0-312.a.1.10 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-312.b.1.12 | $312$ | $2$ | $2$ | $0$ |