Invariants
| Level: | $12$ | $\SL_2$-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 $\quad(D =$ $-8$) |
Other labels
| Cummins and Pauli (CP) label: | 6E0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.12.0.6 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}0&11\\1&0\end{bmatrix}$, $\begin{bmatrix}0&11\\7&9\end{bmatrix}$, $\begin{bmatrix}7&3\\3&2\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $C_2^2:\GL(2,\mathbb{Z}/4)$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 12-isogeny field degree: | $12$ |
| Cyclic 12-torsion field degree: | $48$ |
| Full 12-torsion field degree: | $384$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 11 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{3^3}{2^6}\cdot\frac{(3x+y)^{12}(x^{2}-12y^{2})^{3}(x^{2}+4y^{2})^{3}}{y^{6}x^{6}(3x+y)^{12}}$ |
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 |
|---|---|---|---|---|
| 12.24.1.a.1 | $12$ | $2$ | $2$ | $1$ |
| 12.24.1.b.1 | $12$ | $2$ | $2$ | $1$ |
| 12.24.1.d.1 | $12$ | $2$ | $2$ | $1$ |
| 12.24.1.e.1 | $12$ | $2$ | $2$ | $1$ |
| 12.36.0.d.1 | $12$ | $3$ | $3$ | $0$ |
| 12.48.1.q.1 | $12$ | $4$ | $4$ | $1$ |
| 24.24.1.cb.1 | $24$ | $2$ | $2$ | $1$ |
| 24.24.1.ce.1 | $24$ | $2$ | $2$ | $1$ |
| 24.24.1.ck.1 | $24$ | $2$ | $2$ | $1$ |
| 24.24.1.cq.1 | $24$ | $2$ | $2$ | $1$ |
| 36.36.0.c.1 | $36$ | $3$ | $3$ | $0$ |
| 36.36.1.c.1 | $36$ | $3$ | $3$ | $1$ |
| 36.36.1.d.1 | $36$ | $3$ | $3$ | $1$ |
| 60.24.1.bi.1 | $60$ | $2$ | $2$ | $1$ |
| 60.24.1.bj.1 | $60$ | $2$ | $2$ | $1$ |
| 60.24.1.bl.1 | $60$ | $2$ | $2$ | $1$ |
| 60.24.1.bm.1 | $60$ | $2$ | $2$ | $1$ |
| 60.60.4.cp.1 | $60$ | $5$ | $5$ | $4$ |
| 60.72.3.bct.1 | $60$ | $6$ | $6$ | $3$ |
| 60.120.7.lz.1 | $60$ | $10$ | $10$ | $7$ |
| 84.24.1.q.1 | $84$ | $2$ | $2$ | $1$ |
| 84.24.1.r.1 | $84$ | $2$ | $2$ | $1$ |
| 84.24.1.t.1 | $84$ | $2$ | $2$ | $1$ |
| 84.24.1.u.1 | $84$ | $2$ | $2$ | $1$ |
| 84.96.7.bx.1 | $84$ | $8$ | $8$ | $7$ |
| 84.252.14.b.1 | $84$ | $21$ | $21$ | $14$ |
| 84.336.21.lp.1 | $84$ | $28$ | $28$ | $21$ |
| 120.24.1.oc.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.of.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.oo.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.or.1 | $120$ | $2$ | $2$ | $1$ |
| 132.24.1.q.1 | $132$ | $2$ | $2$ | $1$ |
| 132.24.1.r.1 | $132$ | $2$ | $2$ | $1$ |
| 132.24.1.t.1 | $132$ | $2$ | $2$ | $1$ |
| 132.24.1.u.1 | $132$ | $2$ | $2$ | $1$ |
| 132.144.11.bx.1 | $132$ | $12$ | $12$ | $11$ |
| 156.24.1.q.1 | $156$ | $2$ | $2$ | $1$ |
| 156.24.1.r.1 | $156$ | $2$ | $2$ | $1$ |
| 156.24.1.t.1 | $156$ | $2$ | $2$ | $1$ |
| 156.24.1.u.1 | $156$ | $2$ | $2$ | $1$ |
| 156.168.11.gv.1 | $156$ | $14$ | $14$ | $11$ |
| 168.24.1.lh.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.lk.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.lt.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.lw.1 | $168$ | $2$ | $2$ | $1$ |
| 204.24.1.q.1 | $204$ | $2$ | $2$ | $1$ |
| 204.24.1.r.1 | $204$ | $2$ | $2$ | $1$ |
| 204.24.1.t.1 | $204$ | $2$ | $2$ | $1$ |
| 204.24.1.u.1 | $204$ | $2$ | $2$ | $1$ |
| 204.216.15.fb.1 | $204$ | $18$ | $18$ | $15$ |
| 228.24.1.q.1 | $228$ | $2$ | $2$ | $1$ |
| 228.24.1.r.1 | $228$ | $2$ | $2$ | $1$ |
| 228.24.1.t.1 | $228$ | $2$ | $2$ | $1$ |
| 228.24.1.u.1 | $228$ | $2$ | $2$ | $1$ |
| 228.240.19.bx.1 | $228$ | $20$ | $20$ | $19$ |
| 264.24.1.li.1 | $264$ | $2$ | $2$ | $1$ |
| 264.24.1.ll.1 | $264$ | $2$ | $2$ | $1$ |
| 264.24.1.lu.1 | $264$ | $2$ | $2$ | $1$ |
| 264.24.1.lx.1 | $264$ | $2$ | $2$ | $1$ |
| 276.24.1.q.1 | $276$ | $2$ | $2$ | $1$ |
| 276.24.1.r.1 | $276$ | $2$ | $2$ | $1$ |
| 276.24.1.t.1 | $276$ | $2$ | $2$ | $1$ |
| 276.24.1.u.1 | $276$ | $2$ | $2$ | $1$ |
| 276.288.23.bx.1 | $276$ | $24$ | $24$ | $23$ |
| 312.24.1.li.1 | $312$ | $2$ | $2$ | $1$ |
| 312.24.1.ll.1 | $312$ | $2$ | $2$ | $1$ |
| 312.24.1.lu.1 | $312$ | $2$ | $2$ | $1$ |
| 312.24.1.lx.1 | $312$ | $2$ | $2$ | $1$ |