Invariants
| Level: | $12$ | $\SL_2$-level: | $4$ | ||||
| Index: | $24$ | $\PSL_2$-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: | 12.24.0.8 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}1&4\\6&5\end{bmatrix}$, $\begin{bmatrix}3&10\\8&5\end{bmatrix}$, $\begin{bmatrix}5&10\\10&9\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $C_2^2\times \GL(2,3)$ |
| Contains $-I$: | no $\quad$ (see 12.12.0.b.1 for the level structure with $-I$) |
| Cyclic 12-isogeny field degree: | $8$ |
| Cyclic 12-torsion field degree: | $16$ |
| Full 12-torsion field degree: | $192$ |
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
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 4.12.0-2.a.1.1 | $4$ | $2$ | $2$ | $0$ | $0$ |
| 12.12.0-2.a.1.1 | $12$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 12.48.0-12.b.1.1 | $12$ | $2$ | $2$ | $0$ |
| 12.48.0-12.b.1.2 | $12$ | $2$ | $2$ | $0$ |
| 12.48.0-12.c.1.1 | $12$ | $2$ | $2$ | $0$ |
| 12.48.0-12.c.1.4 | $12$ | $2$ | $2$ | $0$ |
| 12.72.2-12.d.1.3 | $12$ | $3$ | $3$ | $2$ |
| 12.96.1-12.d.1.5 | $12$ | $4$ | $4$ | $1$ |
| 24.48.0-24.d.1.2 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.d.1.3 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.g.1.2 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.g.1.3 | $24$ | $2$ | $2$ | $0$ |
| 36.648.22-36.d.1.2 | $36$ | $27$ | $27$ | $22$ |
| 60.48.0-60.f.1.2 | $60$ | $2$ | $2$ | $0$ |
| 60.48.0-60.f.1.3 | $60$ | $2$ | $2$ | $0$ |
| 60.48.0-60.g.1.7 | $60$ | $2$ | $2$ | $0$ |
| 60.48.0-60.g.1.8 | $60$ | $2$ | $2$ | $0$ |
| 60.120.4-60.b.1.4 | $60$ | $5$ | $5$ | $4$ |
| 60.144.3-60.b.1.7 | $60$ | $6$ | $6$ | $3$ |
| 60.240.7-60.b.1.15 | $60$ | $10$ | $10$ | $7$ |
| 84.48.0-84.f.1.1 | $84$ | $2$ | $2$ | $0$ |
| 84.48.0-84.f.1.8 | $84$ | $2$ | $2$ | $0$ |
| 84.48.0-84.g.1.1 | $84$ | $2$ | $2$ | $0$ |
| 84.48.0-84.g.1.7 | $84$ | $2$ | $2$ | $0$ |
| 84.192.5-84.b.1.5 | $84$ | $8$ | $8$ | $5$ |
| 84.504.16-84.b.1.4 | $84$ | $21$ | $21$ | $16$ |
| 120.48.0-120.n.1.9 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.n.1.12 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.q.1.9 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.q.1.12 | $120$ | $2$ | $2$ | $0$ |
| 132.48.0-132.f.1.1 | $132$ | $2$ | $2$ | $0$ |
| 132.48.0-132.f.1.8 | $132$ | $2$ | $2$ | $0$ |
| 132.48.0-132.g.1.1 | $132$ | $2$ | $2$ | $0$ |
| 132.48.0-132.g.1.7 | $132$ | $2$ | $2$ | $0$ |
| 132.288.9-132.b.1.8 | $132$ | $12$ | $12$ | $9$ |
| 156.48.0-156.f.1.1 | $156$ | $2$ | $2$ | $0$ |
| 156.48.0-156.f.1.8 | $156$ | $2$ | $2$ | $0$ |
| 156.48.0-156.g.1.1 | $156$ | $2$ | $2$ | $0$ |
| 156.48.0-156.g.1.7 | $156$ | $2$ | $2$ | $0$ |
| 156.336.11-156.b.1.2 | $156$ | $14$ | $14$ | $11$ |
| 168.48.0-168.n.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.n.1.15 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.q.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.q.1.15 | $168$ | $2$ | $2$ | $0$ |
| 204.48.0-204.f.1.2 | $204$ | $2$ | $2$ | $0$ |
| 204.48.0-204.f.1.4 | $204$ | $2$ | $2$ | $0$ |
| 204.48.0-204.g.1.7 | $204$ | $2$ | $2$ | $0$ |
| 204.48.0-204.g.1.8 | $204$ | $2$ | $2$ | $0$ |
| 204.432.15-204.b.1.7 | $204$ | $18$ | $18$ | $15$ |
| 228.48.0-228.f.1.1 | $228$ | $2$ | $2$ | $0$ |
| 228.48.0-228.f.1.8 | $228$ | $2$ | $2$ | $0$ |
| 228.48.0-228.g.1.1 | $228$ | $2$ | $2$ | $0$ |
| 228.48.0-228.g.1.7 | $228$ | $2$ | $2$ | $0$ |
| 228.480.17-228.b.1.1 | $228$ | $20$ | $20$ | $17$ |
| 264.48.0-264.n.1.3 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.n.1.6 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.q.1.3 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.q.1.6 | $264$ | $2$ | $2$ | $0$ |
| 276.48.0-276.f.1.1 | $276$ | $2$ | $2$ | $0$ |
| 276.48.0-276.f.1.8 | $276$ | $2$ | $2$ | $0$ |
| 276.48.0-276.g.1.1 | $276$ | $2$ | $2$ | $0$ |
| 276.48.0-276.g.1.7 | $276$ | $2$ | $2$ | $0$ |
| 312.48.0-312.n.1.2 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.n.1.15 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.q.1.2 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.q.1.15 | $312$ | $2$ | $2$ | $0$ |