Invariants
| Level: | $12$ | $\SL_2$-level: | $4$ | ||||
| Index: | $6$ | $\PSL_2$-index: | $6$ | ||||
| Genus: | $0 = 1 + \frac{ 6 }{12} - \frac{ 2 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $2\cdot4$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $2$ 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: | 4C0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.6.0.3 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}1&4\\4&11\end{bmatrix}$, $\begin{bmatrix}9&1\\4&9\end{bmatrix}$, $\begin{bmatrix}11&9\\10&11\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $(C_2\times \GL(2,3)):D_4$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 12-isogeny field degree: | $8$ |
| Cyclic 12-torsion field degree: | $32$ |
| Full 12-torsion field degree: | $768$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 1518 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 6 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^4\cdot3}\cdot\frac{x^{6}(3x^{2}-64y^{2})^{3}}{y^{4}x^{8}}$ |
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 |
|---|---|---|---|---|---|
| $X_0(2)$ | $2$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 12.12.0.b.1 | $12$ | $2$ | $2$ | $0$ |
| 12.12.0.c.1 | $12$ | $2$ | $2$ | $0$ |
| 12.12.0.f.1 | $12$ | $2$ | $2$ | $0$ |
| 12.12.0.g.1 | $12$ | $2$ | $2$ | $0$ |
| 12.18.0.i.1 | $12$ | $3$ | $3$ | $0$ |
| 12.24.1.i.1 | $12$ | $4$ | $4$ | $1$ |
| 24.12.0.e.1 | $24$ | $2$ | $2$ | $0$ |
| 24.12.0.i.1 | $24$ | $2$ | $2$ | $0$ |
| 24.12.0.r.1 | $24$ | $2$ | $2$ | $0$ |
| 24.12.0.u.1 | $24$ | $2$ | $2$ | $0$ |
| 36.162.10.o.1 | $36$ | $27$ | $27$ | $10$ |
| 60.12.0.z.1 | $60$ | $2$ | $2$ | $0$ |
| 60.12.0.ba.1 | $60$ | $2$ | $2$ | $0$ |
| 60.12.0.bd.1 | $60$ | $2$ | $2$ | $0$ |
| 60.12.0.be.1 | $60$ | $2$ | $2$ | $0$ |
| 60.30.2.m.1 | $60$ | $5$ | $5$ | $2$ |
| 60.36.1.dq.1 | $60$ | $6$ | $6$ | $1$ |
| 60.60.3.bu.1 | $60$ | $10$ | $10$ | $3$ |
| 84.12.0.z.1 | $84$ | $2$ | $2$ | $0$ |
| 84.12.0.ba.1 | $84$ | $2$ | $2$ | $0$ |
| 84.12.0.bd.1 | $84$ | $2$ | $2$ | $0$ |
| 84.12.0.be.1 | $84$ | $2$ | $2$ | $0$ |
| 84.48.3.ce.1 | $84$ | $8$ | $8$ | $3$ |
| 84.126.6.e.1 | $84$ | $21$ | $21$ | $6$ |
| 84.168.9.bg.1 | $84$ | $28$ | $28$ | $9$ |
| 120.12.0.dc.1 | $120$ | $2$ | $2$ | $0$ |
| 120.12.0.df.1 | $120$ | $2$ | $2$ | $0$ |
| 120.12.0.do.1 | $120$ | $2$ | $2$ | $0$ |
| 120.12.0.dr.1 | $120$ | $2$ | $2$ | $0$ |
| 132.12.0.z.1 | $132$ | $2$ | $2$ | $0$ |
| 132.12.0.ba.1 | $132$ | $2$ | $2$ | $0$ |
| 132.12.0.bd.1 | $132$ | $2$ | $2$ | $0$ |
| 132.12.0.be.1 | $132$ | $2$ | $2$ | $0$ |
| 132.72.5.e.1 | $132$ | $12$ | $12$ | $5$ |
| 132.330.20.e.1 | $132$ | $55$ | $55$ | $20$ |
| 132.330.22.i.1 | $132$ | $55$ | $55$ | $22$ |
| 156.12.0.z.1 | $156$ | $2$ | $2$ | $0$ |
| 156.12.0.ba.1 | $156$ | $2$ | $2$ | $0$ |
| 156.12.0.bd.1 | $156$ | $2$ | $2$ | $0$ |
| 156.12.0.be.1 | $156$ | $2$ | $2$ | $0$ |
| 156.84.5.o.1 | $156$ | $14$ | $14$ | $5$ |
| 168.12.0.dc.1 | $168$ | $2$ | $2$ | $0$ |
| 168.12.0.df.1 | $168$ | $2$ | $2$ | $0$ |
| 168.12.0.do.1 | $168$ | $2$ | $2$ | $0$ |
| 168.12.0.dr.1 | $168$ | $2$ | $2$ | $0$ |
| 204.12.0.z.1 | $204$ | $2$ | $2$ | $0$ |
| 204.12.0.ba.1 | $204$ | $2$ | $2$ | $0$ |
| 204.12.0.bd.1 | $204$ | $2$ | $2$ | $0$ |
| 204.12.0.be.1 | $204$ | $2$ | $2$ | $0$ |
| 204.108.7.p.1 | $204$ | $18$ | $18$ | $7$ |
| 228.12.0.z.1 | $228$ | $2$ | $2$ | $0$ |
| 228.12.0.ba.1 | $228$ | $2$ | $2$ | $0$ |
| 228.12.0.bd.1 | $228$ | $2$ | $2$ | $0$ |
| 228.12.0.be.1 | $228$ | $2$ | $2$ | $0$ |
| 228.120.9.e.1 | $228$ | $20$ | $20$ | $9$ |
| 264.12.0.dc.1 | $264$ | $2$ | $2$ | $0$ |
| 264.12.0.df.1 | $264$ | $2$ | $2$ | $0$ |
| 264.12.0.do.1 | $264$ | $2$ | $2$ | $0$ |
| 264.12.0.dr.1 | $264$ | $2$ | $2$ | $0$ |
| 276.12.0.z.1 | $276$ | $2$ | $2$ | $0$ |
| 276.12.0.ba.1 | $276$ | $2$ | $2$ | $0$ |
| 276.12.0.bd.1 | $276$ | $2$ | $2$ | $0$ |
| 276.12.0.be.1 | $276$ | $2$ | $2$ | $0$ |
| 276.144.11.e.1 | $276$ | $24$ | $24$ | $11$ |
| 312.12.0.dc.1 | $312$ | $2$ | $2$ | $0$ |
| 312.12.0.df.1 | $312$ | $2$ | $2$ | $0$ |
| 312.12.0.do.1 | $312$ | $2$ | $2$ | $0$ |
| 312.12.0.dr.1 | $312$ | $2$ | $2$ | $0$ |