Invariants
| Level: | $12$ | $\SL_2$-level: | $4$ | ||||
| Index: | $12$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4,-12$) |
Other labels
| Cummins and Pauli (CP) label: | 4E0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.12.0.19 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}7&10\\2&1\end{bmatrix}$, $\begin{bmatrix}9&11\\4&7\end{bmatrix}$, $\begin{bmatrix}11&9\\10&1\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $C_2^3:\GL(2,3)$ |
| 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: | $384$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 70 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\cdot3^3\,\frac{x^{12}(x^{2}+4y^{2})^{3}(9x^{2}+4y^{2})^{3}}{x^{12}(3x^{2}-4y^{2})^{4}(3x^{2}+4y^{2})^{2}}$ |
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 |
|---|---|---|---|---|---|
| 4.6.0.e.1 | $4$ | $2$ | $2$ | $0$ | $0$ |
| 12.6.0.a.1 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 12.6.0.c.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 |
|---|---|---|---|---|
| 12.24.0.m.1 | $12$ | $2$ | $2$ | $0$ |
| 12.24.0.n.1 | $12$ | $2$ | $2$ | $0$ |
| 12.36.2.bx.1 | $12$ | $3$ | $3$ | $2$ |
| 12.48.1.p.1 | $12$ | $4$ | $4$ | $1$ |
| 24.24.0.ci.1 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0.cj.1 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0.cw.1 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0.cx.1 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0.dc.1 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0.dd.1 | $24$ | $2$ | $2$ | $0$ |
| 24.24.1.bc.1 | $24$ | $2$ | $2$ | $1$ |
| 24.24.1.be.1 | $24$ | $2$ | $2$ | $1$ |
| 24.24.1.em.1 | $24$ | $2$ | $2$ | $1$ |
| 24.24.1.eo.1 | $24$ | $2$ | $2$ | $1$ |
| 36.324.22.cn.1 | $36$ | $27$ | $27$ | $22$ |
| 60.24.0.m.1 | $60$ | $2$ | $2$ | $0$ |
| 60.24.0.n.1 | $60$ | $2$ | $2$ | $0$ |
| 60.60.4.bk.1 | $60$ | $5$ | $5$ | $4$ |
| 60.72.3.oy.1 | $60$ | $6$ | $6$ | $3$ |
| 60.120.7.dm.1 | $60$ | $10$ | $10$ | $7$ |
| 84.24.0.k.1 | $84$ | $2$ | $2$ | $0$ |
| 84.24.0.l.1 | $84$ | $2$ | $2$ | $0$ |
| 84.96.5.bs.1 | $84$ | $8$ | $8$ | $5$ |
| 84.252.16.dg.1 | $84$ | $21$ | $21$ | $16$ |
| 84.336.21.dg.1 | $84$ | $28$ | $28$ | $21$ |
| 120.24.0.ey.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.ez.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.fc.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.fd.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.fi.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.fj.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.1.ds.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.dt.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.ia.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.ib.1 | $120$ | $2$ | $2$ | $1$ |
| 132.24.0.k.1 | $132$ | $2$ | $2$ | $0$ |
| 132.24.0.l.1 | $132$ | $2$ | $2$ | $0$ |
| 132.144.9.bk.1 | $132$ | $12$ | $12$ | $9$ |
| 156.24.0.m.1 | $156$ | $2$ | $2$ | $0$ |
| 156.24.0.n.1 | $156$ | $2$ | $2$ | $0$ |
| 156.168.11.ci.1 | $156$ | $14$ | $14$ | $11$ |
| 168.24.0.es.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.et.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.ew.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.ex.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.fc.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.fd.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.1.dg.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.dh.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.gq.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.gr.1 | $168$ | $2$ | $2$ | $1$ |
| 204.24.0.m.1 | $204$ | $2$ | $2$ | $0$ |
| 204.24.0.n.1 | $204$ | $2$ | $2$ | $0$ |
| 204.216.15.ci.1 | $204$ | $18$ | $18$ | $15$ |
| 228.24.0.k.1 | $228$ | $2$ | $2$ | $0$ |
| 228.24.0.l.1 | $228$ | $2$ | $2$ | $0$ |
| 228.240.17.bk.1 | $228$ | $20$ | $20$ | $17$ |
| 264.24.0.es.1 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0.et.1 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0.ew.1 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0.ex.1 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0.fc.1 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0.fd.1 | $264$ | $2$ | $2$ | $0$ |
| 264.24.1.dg.1 | $264$ | $2$ | $2$ | $1$ |
| 264.24.1.dh.1 | $264$ | $2$ | $2$ | $1$ |
| 264.24.1.gq.1 | $264$ | $2$ | $2$ | $1$ |
| 264.24.1.gr.1 | $264$ | $2$ | $2$ | $1$ |
| 276.24.0.k.1 | $276$ | $2$ | $2$ | $0$ |
| 276.24.0.l.1 | $276$ | $2$ | $2$ | $0$ |
| 276.288.21.bk.1 | $276$ | $24$ | $24$ | $21$ |
| 312.24.0.ey.1 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0.ez.1 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0.fc.1 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0.fd.1 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0.fi.1 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0.fj.1 | $312$ | $2$ | $2$ | $0$ |
| 312.24.1.dg.1 | $312$ | $2$ | $2$ | $1$ |
| 312.24.1.dh.1 | $312$ | $2$ | $2$ | $1$ |
| 312.24.1.gq.1 | $312$ | $2$ | $2$ | $1$ |
| 312.24.1.gr.1 | $312$ | $2$ | $2$ | $1$ |