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: | $4$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 4E0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.12.0.15 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}9&10\\10&11\end{bmatrix}$, $\begin{bmatrix}11&5\\0&7\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $C_2^2.\GL(2,\mathbb{Z}/4)$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 24.24.0-12.f.1.1, 24.24.0-12.f.1.2, 24.24.0-12.f.1.3, 24.24.0-12.f.1.4, 120.24.0-12.f.1.1, 120.24.0-12.f.1.2, 120.24.0-12.f.1.3, 120.24.0-12.f.1.4, 168.24.0-12.f.1.1, 168.24.0-12.f.1.2, 168.24.0-12.f.1.3, 168.24.0-12.f.1.4, 264.24.0-12.f.1.1, 264.24.0-12.f.1.2, 264.24.0-12.f.1.3, 264.24.0-12.f.1.4, 312.24.0-12.f.1.1, 312.24.0-12.f.1.2, 312.24.0-12.f.1.3, 312.24.0-12.f.1.4 |
| Cyclic 12-isogeny field degree: | $8$ |
| Cyclic 12-torsion field degree: | $32$ |
| Full 12-torsion field degree: | $384$ |
Models
Smooth plane model Smooth plane model
| $ 0 $ | $=$ | $ 4 x^{2} - 192 y^{2} - 3 z^{2} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 4.6.0.b.1 | $4$ | $2$ | $2$ | $0$ | $0$ |
| 12.6.0.a.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 |
|---|---|---|---|---|
| 12.36.2.r.1 | $12$ | $3$ | $3$ | $2$ |
| 12.48.1.j.1 | $12$ | $4$ | $4$ | $1$ |
| 36.324.22.z.1 | $36$ | $27$ | $27$ | $22$ |
| 60.60.4.j.1 | $60$ | $5$ | $5$ | $4$ |
| 60.72.3.er.1 | $60$ | $6$ | $6$ | $3$ |
| 60.120.7.r.1 | $60$ | $10$ | $10$ | $7$ |
| 84.96.5.j.1 | $84$ | $8$ | $8$ | $5$ |
| 84.252.16.r.1 | $84$ | $21$ | $21$ | $16$ |
| 84.336.21.r.1 | $84$ | $28$ | $28$ | $21$ |
| 132.144.9.j.1 | $132$ | $12$ | $12$ | $9$ |
| 156.168.11.n.1 | $156$ | $14$ | $14$ | $11$ |
| 204.216.15.n.1 | $204$ | $18$ | $18$ | $15$ |
| 228.240.17.j.1 | $228$ | $20$ | $20$ | $17$ |
| 276.288.21.j.1 | $276$ | $24$ | $24$ | $21$ |