Invariants
| Level: | $12$ | $\SL_2$-level: | $6$ | ||||
| Index: | $16$ | $\PSL_2$-index: | $8$ | ||||
| Genus: | $0 = 1 + \frac{ 8 }{12} - \frac{ 0 }{4} - \frac{ 2 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $2\cdot6$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $2$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-12$) |
Other labels
| Cummins and Pauli (CP) label: | 6C0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.16.0.16 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}5&11\\3&4\end{bmatrix}$, $\begin{bmatrix}11&4\\9&5\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $D_6:S_4$ |
| Contains $-I$: | no $\quad$ (see 12.8.0.b.1 for the level structure with $-I$) |
| Cyclic 12-isogeny field degree: | $6$ |
| Cyclic 12-torsion field degree: | $24$ |
| Full 12-torsion field degree: | $288$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 163 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 8 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{3^3}{2^6}\cdot\frac{(3x+y)^{8}(x^{2}+4y^{2})^{3}(x^{2}+36y^{2})}{y^{6}x^{2}(3x+y)^{8}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 6.8.0-3.a.1.2 | $6$ | $2$ | $2$ | $0$ | $0$ |
| 12.8.0-3.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.h.1.3 | $12$ | $3$ | $3$ | $0$ |
| 12.48.1-12.d.1.2 | $12$ | $3$ | $3$ | $1$ |
| 12.64.1-12.d.1.2 | $12$ | $4$ | $4$ | $1$ |
| 36.48.0-36.c.1.3 | $36$ | $3$ | $3$ | $0$ |
| 36.48.1-36.b.1.2 | $36$ | $3$ | $3$ | $1$ |
| 36.48.2-36.b.1.2 | $36$ | $3$ | $3$ | $2$ |
| 60.80.2-60.b.1.3 | $60$ | $5$ | $5$ | $2$ |
| 60.96.3-60.t.1.5 | $60$ | $6$ | $6$ | $3$ |
| 60.160.5-60.b.1.13 | $60$ | $10$ | $10$ | $5$ |
| 84.128.3-84.b.1.1 | $84$ | $8$ | $8$ | $3$ |
| 84.336.12-84.b.1.15 | $84$ | $21$ | $21$ | $12$ |
| 84.448.15-84.b.1.11 | $84$ | $28$ | $28$ | $15$ |
| 132.192.7-132.b.1.11 | $132$ | $12$ | $12$ | $7$ |
| 156.224.7-156.b.1.7 | $156$ | $14$ | $14$ | $7$ |
| 204.288.11-204.b.1.13 | $204$ | $18$ | $18$ | $11$ |
| 228.320.11-228.b.1.9 | $228$ | $20$ | $20$ | $11$ |
| 276.384.15-276.b.1.11 | $276$ | $24$ | $24$ | $15$ |