Invariants
| Level: | $12$ | $\SL_2$-level: | $12$ | ||||
| Index: | $18$ | $\PSL_2$-index: | $18$ | ||||
| Genus: | $0 = 1 + \frac{ 18 }{12} - \frac{ 6 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $6\cdot12$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $6$ 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: | 12C0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.18.0.8 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}5&3\\8&7\end{bmatrix}$, $\begin{bmatrix}7&1\\10&5\end{bmatrix}$, $\begin{bmatrix}9&1\\4&1\end{bmatrix}$, $\begin{bmatrix}9&7\\2&3\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $C_2^2.D_4^2$ |
| 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: | $256$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 4 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 18 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^{12}\cdot3^3}\cdot\frac{(3x+y)^{18}(27x^{6}-1024y^{6})^{3}}{y^{12}x^{6}(3x+y)^{18}}$ |
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 |
|---|---|---|---|---|---|
| 6.9.0.a.1 | $6$ | $2$ | $2$ | $0$ | $0$ |
| 12.6.0.f.1 | $12$ | $3$ | $3$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 12.36.0.e.1 | $12$ | $2$ | $2$ | $0$ |
| 12.36.0.f.1 | $12$ | $2$ | $2$ | $0$ |
| 12.36.0.i.1 | $12$ | $2$ | $2$ | $0$ |
| 12.36.0.j.1 | $12$ | $2$ | $2$ | $0$ |
| 12.36.1.bc.1 | $12$ | $2$ | $2$ | $1$ |
| 12.36.1.bd.1 | $12$ | $2$ | $2$ | $1$ |
| 12.36.1.bg.1 | $12$ | $2$ | $2$ | $1$ |
| 12.36.1.bh.1 | $12$ | $2$ | $2$ | $1$ |
| 12.36.2.d.1 | $12$ | $2$ | $2$ | $2$ |
| 12.36.2.j.1 | $12$ | $2$ | $2$ | $2$ |
| 12.36.2.r.1 | $12$ | $2$ | $2$ | $2$ |
| 12.36.2.s.1 | $12$ | $2$ | $2$ | $2$ |
| 24.36.0.i.1 | $24$ | $2$ | $2$ | $0$ |
| 24.36.0.l.1 | $24$ | $2$ | $2$ | $0$ |
| 24.36.0.u.1 | $24$ | $2$ | $2$ | $0$ |
| 24.36.0.x.1 | $24$ | $2$ | $2$ | $0$ |
| 24.36.1.du.1 | $24$ | $2$ | $2$ | $1$ |
| 24.36.1.dx.1 | $24$ | $2$ | $2$ | $1$ |
| 24.36.1.eg.1 | $24$ | $2$ | $2$ | $1$ |
| 24.36.1.ej.1 | $24$ | $2$ | $2$ | $1$ |
| 24.36.2.u.1 | $24$ | $2$ | $2$ | $2$ |
| 24.36.2.bd.1 | $24$ | $2$ | $2$ | $2$ |
| 24.36.2.cb.1 | $24$ | $2$ | $2$ | $2$ |
| 24.36.2.ce.1 | $24$ | $2$ | $2$ | $2$ |
| 36.54.2.g.1 | $36$ | $3$ | $3$ | $2$ |
| 36.162.8.c.1 | $36$ | $9$ | $9$ | $8$ |
| 60.36.0.bq.1 | $60$ | $2$ | $2$ | $0$ |
| 60.36.0.br.1 | $60$ | $2$ | $2$ | $0$ |
| 60.36.0.bu.1 | $60$ | $2$ | $2$ | $0$ |
| 60.36.0.bv.1 | $60$ | $2$ | $2$ | $0$ |
| 60.36.1.fa.1 | $60$ | $2$ | $2$ | $1$ |
| 60.36.1.fb.1 | $60$ | $2$ | $2$ | $1$ |
| 60.36.1.fe.1 | $60$ | $2$ | $2$ | $1$ |
| 60.36.1.ff.1 | $60$ | $2$ | $2$ | $1$ |
| 60.36.2.fa.1 | $60$ | $2$ | $2$ | $2$ |
| 60.36.2.fb.1 | $60$ | $2$ | $2$ | $2$ |
| 60.36.2.fe.1 | $60$ | $2$ | $2$ | $2$ |
| 60.36.2.ff.1 | $60$ | $2$ | $2$ | $2$ |
| 60.90.6.u.1 | $60$ | $5$ | $5$ | $6$ |
| 60.108.5.u.1 | $60$ | $6$ | $6$ | $5$ |
| 60.180.11.em.1 | $60$ | $10$ | $10$ | $11$ |
| 84.36.0.bk.1 | $84$ | $2$ | $2$ | $0$ |
| 84.36.0.bl.1 | $84$ | $2$ | $2$ | $0$ |
| 84.36.0.bo.1 | $84$ | $2$ | $2$ | $0$ |
| 84.36.0.bp.1 | $84$ | $2$ | $2$ | $0$ |
| 84.36.1.eq.1 | $84$ | $2$ | $2$ | $1$ |
| 84.36.1.er.1 | $84$ | $2$ | $2$ | $1$ |
| 84.36.1.eu.1 | $84$ | $2$ | $2$ | $1$ |
| 84.36.1.ev.1 | $84$ | $2$ | $2$ | $1$ |
| 84.36.2.fa.1 | $84$ | $2$ | $2$ | $2$ |
| 84.36.2.fb.1 | $84$ | $2$ | $2$ | $2$ |
| 84.36.2.fe.1 | $84$ | $2$ | $2$ | $2$ |
| 84.36.2.ff.1 | $84$ | $2$ | $2$ | $2$ |
| 84.144.11.m.1 | $84$ | $8$ | $8$ | $11$ |
| 84.378.22.u.1 | $84$ | $21$ | $21$ | $22$ |
| 120.36.0.ek.1 | $120$ | $2$ | $2$ | $0$ |
| 120.36.0.en.1 | $120$ | $2$ | $2$ | $0$ |
| 120.36.0.ew.1 | $120$ | $2$ | $2$ | $0$ |
| 120.36.0.ez.1 | $120$ | $2$ | $2$ | $0$ |
| 120.36.1.pq.1 | $120$ | $2$ | $2$ | $1$ |
| 120.36.1.pt.1 | $120$ | $2$ | $2$ | $1$ |
| 120.36.1.qc.1 | $120$ | $2$ | $2$ | $1$ |
| 120.36.1.qf.1 | $120$ | $2$ | $2$ | $1$ |
| 120.36.2.po.1 | $120$ | $2$ | $2$ | $2$ |
| 120.36.2.pr.1 | $120$ | $2$ | $2$ | $2$ |
| 120.36.2.qa.1 | $120$ | $2$ | $2$ | $2$ |
| 120.36.2.qd.1 | $120$ | $2$ | $2$ | $2$ |
| 132.36.0.bk.1 | $132$ | $2$ | $2$ | $0$ |
| 132.36.0.bl.1 | $132$ | $2$ | $2$ | $0$ |
| 132.36.0.bo.1 | $132$ | $2$ | $2$ | $0$ |
| 132.36.0.bp.1 | $132$ | $2$ | $2$ | $0$ |
| 132.36.1.em.1 | $132$ | $2$ | $2$ | $1$ |
| 132.36.1.en.1 | $132$ | $2$ | $2$ | $1$ |
| 132.36.1.eq.1 | $132$ | $2$ | $2$ | $1$ |
| 132.36.1.er.1 | $132$ | $2$ | $2$ | $1$ |
| 132.36.2.fa.1 | $132$ | $2$ | $2$ | $2$ |
| 132.36.2.fb.1 | $132$ | $2$ | $2$ | $2$ |
| 132.36.2.fe.1 | $132$ | $2$ | $2$ | $2$ |
| 132.36.2.ff.1 | $132$ | $2$ | $2$ | $2$ |
| 132.216.17.m.1 | $132$ | $12$ | $12$ | $17$ |
| 156.36.0.bk.1 | $156$ | $2$ | $2$ | $0$ |
| 156.36.0.bl.1 | $156$ | $2$ | $2$ | $0$ |
| 156.36.0.bo.1 | $156$ | $2$ | $2$ | $0$ |
| 156.36.0.bp.1 | $156$ | $2$ | $2$ | $0$ |
| 156.36.1.eq.1 | $156$ | $2$ | $2$ | $1$ |
| 156.36.1.er.1 | $156$ | $2$ | $2$ | $1$ |
| 156.36.1.eu.1 | $156$ | $2$ | $2$ | $1$ |
| 156.36.1.ev.1 | $156$ | $2$ | $2$ | $1$ |
| 156.36.2.fa.1 | $156$ | $2$ | $2$ | $2$ |
| 156.36.2.fb.1 | $156$ | $2$ | $2$ | $2$ |
| 156.36.2.fe.1 | $156$ | $2$ | $2$ | $2$ |
| 156.36.2.ff.1 | $156$ | $2$ | $2$ | $2$ |
| 156.252.17.u.1 | $156$ | $14$ | $14$ | $17$ |
| 168.36.0.ea.1 | $168$ | $2$ | $2$ | $0$ |
| 168.36.0.ed.1 | $168$ | $2$ | $2$ | $0$ |
| 168.36.0.em.1 | $168$ | $2$ | $2$ | $0$ |
| 168.36.0.ep.1 | $168$ | $2$ | $2$ | $0$ |
| 168.36.1.pe.1 | $168$ | $2$ | $2$ | $1$ |
| 168.36.1.ph.1 | $168$ | $2$ | $2$ | $1$ |
| 168.36.1.pq.1 | $168$ | $2$ | $2$ | $1$ |
| 168.36.1.pt.1 | $168$ | $2$ | $2$ | $1$ |
| 168.36.2.pn.1 | $168$ | $2$ | $2$ | $2$ |
| 168.36.2.pq.1 | $168$ | $2$ | $2$ | $2$ |
| 168.36.2.pz.1 | $168$ | $2$ | $2$ | $2$ |
| 168.36.2.qc.1 | $168$ | $2$ | $2$ | $2$ |
| 204.36.0.bk.1 | $204$ | $2$ | $2$ | $0$ |
| 204.36.0.bl.1 | $204$ | $2$ | $2$ | $0$ |
| 204.36.0.bo.1 | $204$ | $2$ | $2$ | $0$ |
| 204.36.0.bp.1 | $204$ | $2$ | $2$ | $0$ |
| 204.36.1.en.1 | $204$ | $2$ | $2$ | $1$ |
| 204.36.1.eo.1 | $204$ | $2$ | $2$ | $1$ |
| 204.36.1.er.1 | $204$ | $2$ | $2$ | $1$ |
| 204.36.1.es.1 | $204$ | $2$ | $2$ | $1$ |
| 204.36.2.fa.1 | $204$ | $2$ | $2$ | $2$ |
| 204.36.2.fb.1 | $204$ | $2$ | $2$ | $2$ |
| 204.36.2.fe.1 | $204$ | $2$ | $2$ | $2$ |
| 204.36.2.ff.1 | $204$ | $2$ | $2$ | $2$ |
| 204.324.23.bg.1 | $204$ | $18$ | $18$ | $23$ |
| 228.36.0.bk.1 | $228$ | $2$ | $2$ | $0$ |
| 228.36.0.bl.1 | $228$ | $2$ | $2$ | $0$ |
| 228.36.0.bo.1 | $228$ | $2$ | $2$ | $0$ |
| 228.36.0.bp.1 | $228$ | $2$ | $2$ | $0$ |
| 228.36.1.eq.1 | $228$ | $2$ | $2$ | $1$ |
| 228.36.1.er.1 | $228$ | $2$ | $2$ | $1$ |
| 228.36.1.eu.1 | $228$ | $2$ | $2$ | $1$ |
| 228.36.1.ev.1 | $228$ | $2$ | $2$ | $1$ |
| 228.36.2.fa.1 | $228$ | $2$ | $2$ | $2$ |
| 228.36.2.fb.1 | $228$ | $2$ | $2$ | $2$ |
| 228.36.2.fe.1 | $228$ | $2$ | $2$ | $2$ |
| 228.36.2.ff.1 | $228$ | $2$ | $2$ | $2$ |
| 264.36.0.ea.1 | $264$ | $2$ | $2$ | $0$ |
| 264.36.0.ed.1 | $264$ | $2$ | $2$ | $0$ |
| 264.36.0.em.1 | $264$ | $2$ | $2$ | $0$ |
| 264.36.0.ep.1 | $264$ | $2$ | $2$ | $0$ |
| 264.36.1.ov.1 | $264$ | $2$ | $2$ | $1$ |
| 264.36.1.oy.1 | $264$ | $2$ | $2$ | $1$ |
| 264.36.1.ph.1 | $264$ | $2$ | $2$ | $1$ |
| 264.36.1.pk.1 | $264$ | $2$ | $2$ | $1$ |
| 264.36.2.po.1 | $264$ | $2$ | $2$ | $2$ |
| 264.36.2.pr.1 | $264$ | $2$ | $2$ | $2$ |
| 264.36.2.qa.1 | $264$ | $2$ | $2$ | $2$ |
| 264.36.2.qd.1 | $264$ | $2$ | $2$ | $2$ |
| 276.36.0.bk.1 | $276$ | $2$ | $2$ | $0$ |
| 276.36.0.bl.1 | $276$ | $2$ | $2$ | $0$ |
| 276.36.0.bo.1 | $276$ | $2$ | $2$ | $0$ |
| 276.36.0.bp.1 | $276$ | $2$ | $2$ | $0$ |
| 276.36.1.em.1 | $276$ | $2$ | $2$ | $1$ |
| 276.36.1.en.1 | $276$ | $2$ | $2$ | $1$ |
| 276.36.1.eq.1 | $276$ | $2$ | $2$ | $1$ |
| 276.36.1.er.1 | $276$ | $2$ | $2$ | $1$ |
| 276.36.2.fa.1 | $276$ | $2$ | $2$ | $2$ |
| 276.36.2.fb.1 | $276$ | $2$ | $2$ | $2$ |
| 276.36.2.fe.1 | $276$ | $2$ | $2$ | $2$ |
| 276.36.2.ff.1 | $276$ | $2$ | $2$ | $2$ |
| 312.36.0.ea.1 | $312$ | $2$ | $2$ | $0$ |
| 312.36.0.ed.1 | $312$ | $2$ | $2$ | $0$ |
| 312.36.0.em.1 | $312$ | $2$ | $2$ | $0$ |
| 312.36.0.ep.1 | $312$ | $2$ | $2$ | $0$ |
| 312.36.1.pe.1 | $312$ | $2$ | $2$ | $1$ |
| 312.36.1.ph.1 | $312$ | $2$ | $2$ | $1$ |
| 312.36.1.pq.1 | $312$ | $2$ | $2$ | $1$ |
| 312.36.1.pt.1 | $312$ | $2$ | $2$ | $1$ |
| 312.36.2.po.1 | $312$ | $2$ | $2$ | $2$ |
| 312.36.2.pr.1 | $312$ | $2$ | $2$ | $2$ |
| 312.36.2.qa.1 | $312$ | $2$ | $2$ | $2$ |
| 312.36.2.qd.1 | $312$ | $2$ | $2$ | $2$ |