Invariants
| Level: | $6$ | $\SL_2$-level: | $6$ | ||||
| Index: | $12$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $6^{2}$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $4$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-3,-11$) |
Other labels
| Cummins and Pauli (CP) label: | 6E0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 6.12.0.2 |
Level structure
| $\GL_2(\Z/6\Z)$-generators: | $\begin{bmatrix}1&3\\3&2\end{bmatrix}$, $\begin{bmatrix}3&5\\2&3\end{bmatrix}$ |
| $\GL_2(\Z/6\Z)$-subgroup: | $C_3:D_4$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 6-isogeny field degree: | $6$ |
| Cyclic 6-torsion field degree: | $12$ |
| Full 6-torsion field degree: | $24$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 16 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{3^3}{2^6}\cdot\frac{x^{12}(x-2y)^{3}(x+2y)^{3}(x^{2}+12y^{2})^{3}}{y^{6}x^{18}}$ |
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 |
|---|---|---|---|---|---|
| $X_{\mathrm{sp}}^+(3)$ | $3$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 6.24.1.a.1 | $6$ | $2$ | $2$ | $1$ |
| 6.24.1.b.1 | $6$ | $2$ | $2$ | $1$ |
| $X_{\mathrm{sp}}^+(6)$ | $6$ | $3$ | $3$ | $0$ |
| 12.24.1.c.1 | $12$ | $2$ | $2$ | $1$ |
| 12.24.1.f.1 | $12$ | $2$ | $2$ | $1$ |
| 12.48.1.r.1 | $12$ | $4$ | $4$ | $1$ |
| 18.36.0.b.1 | $18$ | $3$ | $3$ | $0$ |
| 18.36.1.c.1 | $18$ | $3$ | $3$ | $1$ |
| 18.36.1.d.1 | $18$ | $3$ | $3$ | $1$ |
| 18.36.1.e.1 | $18$ | $3$ | $3$ | $1$ |
| 24.24.1.by.1 | $24$ | $2$ | $2$ | $1$ |
| 24.24.1.ch.1 | $24$ | $2$ | $2$ | $1$ |
| 24.24.1.cn.1 | $24$ | $2$ | $2$ | $1$ |
| 24.24.1.ct.1 | $24$ | $2$ | $2$ | $1$ |
| 30.24.1.k.1 | $30$ | $2$ | $2$ | $1$ |
| 30.24.1.l.1 | $30$ | $2$ | $2$ | $1$ |
| 30.60.4.h.1 | $30$ | $5$ | $5$ | $4$ |
| 30.72.3.k.1 | $30$ | $6$ | $6$ | $3$ |
| 30.120.7.bb.1 | $30$ | $10$ | $10$ | $7$ |
| 42.24.1.e.1 | $42$ | $2$ | $2$ | $1$ |
| 42.24.1.f.1 | $42$ | $2$ | $2$ | $1$ |
| 42.96.7.r.1 | $42$ | $8$ | $8$ | $7$ |
| 42.252.14.b.1 | $42$ | $21$ | $21$ | $14$ |
| 42.336.21.s.1 | $42$ | $28$ | $28$ | $21$ |
| 60.24.1.bk.1 | $60$ | $2$ | $2$ | $1$ |
| 60.24.1.bn.1 | $60$ | $2$ | $2$ | $1$ |
| 66.24.1.e.1 | $66$ | $2$ | $2$ | $1$ |
| 66.24.1.f.1 | $66$ | $2$ | $2$ | $1$ |
| 66.144.11.r.1 | $66$ | $12$ | $12$ | $11$ |
| 66.660.44.b.1 | $66$ | $55$ | $55$ | $44$ |
| 66.660.48.d.1 | $66$ | $55$ | $55$ | $48$ |
| 66.792.55.s.1 | $66$ | $66$ | $66$ | $55$ |
| 78.24.1.e.1 | $78$ | $2$ | $2$ | $1$ |
| 78.24.1.f.1 | $78$ | $2$ | $2$ | $1$ |
| 78.168.11.k.1 | $78$ | $14$ | $14$ | $11$ |
| 84.24.1.s.1 | $84$ | $2$ | $2$ | $1$ |
| 84.24.1.v.1 | $84$ | $2$ | $2$ | $1$ |
| 102.24.1.e.1 | $102$ | $2$ | $2$ | $1$ |
| 102.24.1.f.1 | $102$ | $2$ | $2$ | $1$ |
| 102.216.15.d.1 | $102$ | $18$ | $18$ | $15$ |
| 114.24.1.e.1 | $114$ | $2$ | $2$ | $1$ |
| 114.24.1.f.1 | $114$ | $2$ | $2$ | $1$ |
| 114.240.19.r.1 | $114$ | $20$ | $20$ | $19$ |
| 120.24.1.oi.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.ol.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.ou.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.ox.1 | $120$ | $2$ | $2$ | $1$ |
| 132.24.1.s.1 | $132$ | $2$ | $2$ | $1$ |
| 132.24.1.v.1 | $132$ | $2$ | $2$ | $1$ |
| 138.24.1.e.1 | $138$ | $2$ | $2$ | $1$ |
| 138.24.1.f.1 | $138$ | $2$ | $2$ | $1$ |
| 138.288.23.r.1 | $138$ | $24$ | $24$ | $23$ |
| 156.24.1.s.1 | $156$ | $2$ | $2$ | $1$ |
| 156.24.1.v.1 | $156$ | $2$ | $2$ | $1$ |
| 168.24.1.ln.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.lq.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.lz.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.mc.1 | $168$ | $2$ | $2$ | $1$ |
| 174.24.1.e.1 | $174$ | $2$ | $2$ | $1$ |
| 174.24.1.f.1 | $174$ | $2$ | $2$ | $1$ |
| 186.24.1.e.1 | $186$ | $2$ | $2$ | $1$ |
| 186.24.1.f.1 | $186$ | $2$ | $2$ | $1$ |
| 204.24.1.s.1 | $204$ | $2$ | $2$ | $1$ |
| 204.24.1.v.1 | $204$ | $2$ | $2$ | $1$ |
| 210.24.1.bi.1 | $210$ | $2$ | $2$ | $1$ |
| 210.24.1.bj.1 | $210$ | $2$ | $2$ | $1$ |
| 222.24.1.e.1 | $222$ | $2$ | $2$ | $1$ |
| 222.24.1.f.1 | $222$ | $2$ | $2$ | $1$ |
| 228.24.1.s.1 | $228$ | $2$ | $2$ | $1$ |
| 228.24.1.v.1 | $228$ | $2$ | $2$ | $1$ |
| 246.24.1.e.1 | $246$ | $2$ | $2$ | $1$ |
| 246.24.1.f.1 | $246$ | $2$ | $2$ | $1$ |
| 258.24.1.e.1 | $258$ | $2$ | $2$ | $1$ |
| 258.24.1.f.1 | $258$ | $2$ | $2$ | $1$ |
| 264.24.1.lo.1 | $264$ | $2$ | $2$ | $1$ |
| 264.24.1.lr.1 | $264$ | $2$ | $2$ | $1$ |
| 264.24.1.ma.1 | $264$ | $2$ | $2$ | $1$ |
| 264.24.1.md.1 | $264$ | $2$ | $2$ | $1$ |
| 276.24.1.s.1 | $276$ | $2$ | $2$ | $1$ |
| 276.24.1.v.1 | $276$ | $2$ | $2$ | $1$ |
| 282.24.1.e.1 | $282$ | $2$ | $2$ | $1$ |
| 282.24.1.f.1 | $282$ | $2$ | $2$ | $1$ |
| 312.24.1.lo.1 | $312$ | $2$ | $2$ | $1$ |
| 312.24.1.lr.1 | $312$ | $2$ | $2$ | $1$ |
| 312.24.1.ma.1 | $312$ | $2$ | $2$ | $1$ |
| 312.24.1.md.1 | $312$ | $2$ | $2$ | $1$ |
| 318.24.1.e.1 | $318$ | $2$ | $2$ | $1$ |
| 318.24.1.f.1 | $318$ | $2$ | $2$ | $1$ |
| 330.24.1.bi.1 | $330$ | $2$ | $2$ | $1$ |
| 330.24.1.bj.1 | $330$ | $2$ | $2$ | $1$ |