Invariants
| Level: | $12$ | $\SL_2$-level: | $4$ | ||||
| Index: | $12$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| 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: | 4E0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.12.0.4 |
Level structure
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 621 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^4\cdot3^2}\cdot\frac{x^{12}(9x^{4}+192x^{2}y^{2}+4096y^{4})^{3}}{y^{4}x^{16}(3x^{2}+64y^{2})^{2}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X(2)$ | $2$ | $2$ | $2$ | $0$ | $0$ |
| 12.6.0.c.1 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 12.6.0.f.1 | $12$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 12.24.0.b.1 | $12$ | $2$ | $2$ | $0$ |
| 12.24.0.c.1 | $12$ | $2$ | $2$ | $0$ |
| 12.36.2.d.1 | $12$ | $3$ | $3$ | $2$ |
| 12.48.1.d.1 | $12$ | $4$ | $4$ | $1$ |
| 24.24.0.d.1 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0.g.1 | $24$ | $2$ | $2$ | $0$ |
| 36.324.22.d.1 | $36$ | $27$ | $27$ | $22$ |
| 60.24.0.f.1 | $60$ | $2$ | $2$ | $0$ |
| 60.24.0.g.1 | $60$ | $2$ | $2$ | $0$ |
| 60.60.4.b.1 | $60$ | $5$ | $5$ | $4$ |
| 60.72.3.b.1 | $60$ | $6$ | $6$ | $3$ |
| 60.120.7.b.1 | $60$ | $10$ | $10$ | $7$ |
| 84.24.0.f.1 | $84$ | $2$ | $2$ | $0$ |
| 84.24.0.g.1 | $84$ | $2$ | $2$ | $0$ |
| 84.96.5.b.1 | $84$ | $8$ | $8$ | $5$ |
| 84.252.16.b.1 | $84$ | $21$ | $21$ | $16$ |
| 84.336.21.b.1 | $84$ | $28$ | $28$ | $21$ |
| 120.24.0.n.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.q.1 | $120$ | $2$ | $2$ | $0$ |
| 132.24.0.f.1 | $132$ | $2$ | $2$ | $0$ |
| 132.24.0.g.1 | $132$ | $2$ | $2$ | $0$ |
| 132.144.9.b.1 | $132$ | $12$ | $12$ | $9$ |
| 156.24.0.f.1 | $156$ | $2$ | $2$ | $0$ |
| 156.24.0.g.1 | $156$ | $2$ | $2$ | $0$ |
| 156.168.11.b.1 | $156$ | $14$ | $14$ | $11$ |
| 168.24.0.n.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.q.1 | $168$ | $2$ | $2$ | $0$ |
| 204.24.0.f.1 | $204$ | $2$ | $2$ | $0$ |
| 204.24.0.g.1 | $204$ | $2$ | $2$ | $0$ |
| 204.216.15.b.1 | $204$ | $18$ | $18$ | $15$ |
| 228.24.0.f.1 | $228$ | $2$ | $2$ | $0$ |
| 228.24.0.g.1 | $228$ | $2$ | $2$ | $0$ |
| 228.240.17.b.1 | $228$ | $20$ | $20$ | $17$ |
| 264.24.0.n.1 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0.q.1 | $264$ | $2$ | $2$ | $0$ |
| 276.24.0.f.1 | $276$ | $2$ | $2$ | $0$ |
| 276.24.0.g.1 | $276$ | $2$ | $2$ | $0$ |
| 276.288.21.b.1 | $276$ | $24$ | $24$ | $21$ |
| 312.24.0.n.1 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0.q.1 | $312$ | $2$ | $2$ | $0$ |