Invariants
| Level: | $12$ | $\SL_2$-level: | $12$ | ||||
| Index: | $12$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 6 }{4} - \frac{ 0 }{3} - \frac{ 1 }{2}$ | ||||||
| Cusps: | $1$ (which is rational) | Cusp widths | $12$ | Cusp orbits | $1$ | ||
| Elliptic points: | $6$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $1$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-3,-4,-11,-19,-43,-67,-163$) |
Other labels
| Cummins and Pauli (CP) label: | 12A0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.12.0.5 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}1&8\\11&11\end{bmatrix}$, $\begin{bmatrix}4&7\\7&5\end{bmatrix}$, $\begin{bmatrix}8&11\\7&5\end{bmatrix}$, $\begin{bmatrix}10&7\\7&11\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $C_6.D_4^2$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 12-isogeny field degree: | $24$ |
| Cyclic 12-torsion field degree: | $96$ |
| 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 13 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^6}\cdot\frac{x^{15}(x^{3}-32y^{3})^{3}}{y^{12}x^{12}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $4$ | $4$ | $0$ | $0$ |
| $X_{\mathrm{ns}}^+(4)$ | $4$ | $3$ | $3$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $4$ | $4$ | $0$ | $0$ |
| $X_{\mathrm{ns}}^+(4)$ | $4$ | $3$ | $3$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 12.24.0.o.1 | $12$ | $2$ | $2$ | $0$ |
| 12.24.0.p.1 | $12$ | $2$ | $2$ | $0$ |
| 12.24.0.q.1 | $12$ | $2$ | $2$ | $0$ |
| $X_{\mathrm{ns}}^+(12)$ | $12$ | $2$ | $2$ | $0$ |
| 12.24.1.m.1 | $12$ | $2$ | $2$ | $1$ |
| 12.24.1.n.1 | $12$ | $2$ | $2$ | $1$ |
| 12.24.1.o.1 | $12$ | $2$ | $2$ | $1$ |
| 12.24.1.p.1 | $12$ | $2$ | $2$ | $1$ |
| 12.24.2.a.1 | $12$ | $2$ | $2$ | $2$ |
| 12.24.2.b.1 | $12$ | $2$ | $2$ | $2$ |
| 12.24.2.c.1 | $12$ | $2$ | $2$ | $2$ |
| 12.24.2.d.1 | $12$ | $2$ | $2$ | $2$ |
| 12.36.1.bs.1 | $12$ | $3$ | $3$ | $1$ |
| 24.24.0.fe.1 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0.ff.1 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0.fg.1 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0.fh.1 | $24$ | $2$ | $2$ | $0$ |
| 24.24.1.eu.1 | $24$ | $2$ | $2$ | $1$ |
| 24.24.1.ev.1 | $24$ | $2$ | $2$ | $1$ |
| 24.24.1.ew.1 | $24$ | $2$ | $2$ | $1$ |
| 24.24.1.ex.1 | $24$ | $2$ | $2$ | $1$ |
| 24.24.2.a.1 | $24$ | $2$ | $2$ | $2$ |
| 24.24.2.b.1 | $24$ | $2$ | $2$ | $2$ |
| 24.24.2.c.1 | $24$ | $2$ | $2$ | $2$ |
| 24.24.2.d.1 | $24$ | $2$ | $2$ | $2$ |
| 24.48.1.mk.1 | $24$ | $4$ | $4$ | $1$ |
| 36.36.1.e.1 | $36$ | $3$ | $3$ | $1$ |
| 36.108.5.y.1 | $36$ | $9$ | $9$ | $5$ |
| 60.24.0.bg.1 | $60$ | $2$ | $2$ | $0$ |
| 60.24.0.bh.1 | $60$ | $2$ | $2$ | $0$ |
| 60.24.0.bi.1 | $60$ | $2$ | $2$ | $0$ |
| 60.24.0.bj.1 | $60$ | $2$ | $2$ | $0$ |
| 60.24.1.bo.1 | $60$ | $2$ | $2$ | $1$ |
| 60.24.1.bp.1 | $60$ | $2$ | $2$ | $1$ |
| 60.24.1.bq.1 | $60$ | $2$ | $2$ | $1$ |
| 60.24.1.br.1 | $60$ | $2$ | $2$ | $1$ |
| 60.24.2.a.1 | $60$ | $2$ | $2$ | $2$ |
| 60.24.2.b.1 | $60$ | $2$ | $2$ | $2$ |
| 60.24.2.c.1 | $60$ | $2$ | $2$ | $2$ |
| 60.24.2.d.1 | $60$ | $2$ | $2$ | $2$ |
| 60.60.4.cq.1 | $60$ | $5$ | $5$ | $4$ |
| 60.72.3.bcu.1 | $60$ | $6$ | $6$ | $3$ |
| 60.120.7.ma.1 | $60$ | $10$ | $10$ | $7$ |
| 84.24.0.be.1 | $84$ | $2$ | $2$ | $0$ |
| 84.24.0.bf.1 | $84$ | $2$ | $2$ | $0$ |
| 84.24.0.bg.1 | $84$ | $2$ | $2$ | $0$ |
| 84.24.0.bh.1 | $84$ | $2$ | $2$ | $0$ |
| 84.24.1.w.1 | $84$ | $2$ | $2$ | $1$ |
| 84.24.1.x.1 | $84$ | $2$ | $2$ | $1$ |
| 84.24.1.y.1 | $84$ | $2$ | $2$ | $1$ |
| 84.24.1.z.1 | $84$ | $2$ | $2$ | $1$ |
| 84.24.2.a.1 | $84$ | $2$ | $2$ | $2$ |
| 84.24.2.b.1 | $84$ | $2$ | $2$ | $2$ |
| 84.24.2.c.1 | $84$ | $2$ | $2$ | $2$ |
| 84.24.2.d.1 | $84$ | $2$ | $2$ | $2$ |
| 84.96.8.i.1 | $84$ | $8$ | $8$ | $8$ |
| 84.252.13.dn.1 | $84$ | $21$ | $21$ | $13$ |
| 84.336.21.lq.1 | $84$ | $28$ | $28$ | $21$ |
| 120.24.0.oq.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.or.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.os.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.ot.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.1.qu.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.qv.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.qw.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.qx.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.2.a.1 | $120$ | $2$ | $2$ | $2$ |
| 120.24.2.b.1 | $120$ | $2$ | $2$ | $2$ |
| 120.24.2.c.1 | $120$ | $2$ | $2$ | $2$ |
| 120.24.2.d.1 | $120$ | $2$ | $2$ | $2$ |
| 132.24.0.be.1 | $132$ | $2$ | $2$ | $0$ |
| 132.24.0.bf.1 | $132$ | $2$ | $2$ | $0$ |
| 132.24.0.bg.1 | $132$ | $2$ | $2$ | $0$ |
| 132.24.0.bh.1 | $132$ | $2$ | $2$ | $0$ |
| 132.24.1.w.1 | $132$ | $2$ | $2$ | $1$ |
| 132.24.1.x.1 | $132$ | $2$ | $2$ | $1$ |
| 132.24.1.y.1 | $132$ | $2$ | $2$ | $1$ |
| 132.24.1.z.1 | $132$ | $2$ | $2$ | $1$ |
| 132.24.2.c.1 | $132$ | $2$ | $2$ | $2$ |
| 132.24.2.d.1 | $132$ | $2$ | $2$ | $2$ |
| 132.24.2.e.1 | $132$ | $2$ | $2$ | $2$ |
| 132.24.2.f.1 | $132$ | $2$ | $2$ | $2$ |
| 132.144.12.i.1 | $132$ | $12$ | $12$ | $12$ |
| 156.24.0.bg.1 | $156$ | $2$ | $2$ | $0$ |
| 156.24.0.bh.1 | $156$ | $2$ | $2$ | $0$ |
| 156.24.0.bi.1 | $156$ | $2$ | $2$ | $0$ |
| 156.24.0.bj.1 | $156$ | $2$ | $2$ | $0$ |
| 156.24.1.w.1 | $156$ | $2$ | $2$ | $1$ |
| 156.24.1.x.1 | $156$ | $2$ | $2$ | $1$ |
| 156.24.1.y.1 | $156$ | $2$ | $2$ | $1$ |
| 156.24.1.z.1 | $156$ | $2$ | $2$ | $1$ |
| 156.24.2.a.1 | $156$ | $2$ | $2$ | $2$ |
| 156.24.2.b.1 | $156$ | $2$ | $2$ | $2$ |
| 156.24.2.c.1 | $156$ | $2$ | $2$ | $2$ |
| 156.24.2.d.1 | $156$ | $2$ | $2$ | $2$ |
| 156.168.11.gw.1 | $156$ | $14$ | $14$ | $11$ |
| 168.24.0.ok.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.ol.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.om.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.on.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.1.nz.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.oa.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.ob.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.oc.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.2.a.1 | $168$ | $2$ | $2$ | $2$ |
| 168.24.2.b.1 | $168$ | $2$ | $2$ | $2$ |
| 168.24.2.c.1 | $168$ | $2$ | $2$ | $2$ |
| 168.24.2.d.1 | $168$ | $2$ | $2$ | $2$ |
| 204.24.0.bg.1 | $204$ | $2$ | $2$ | $0$ |
| 204.24.0.bh.1 | $204$ | $2$ | $2$ | $0$ |
| 204.24.0.bi.1 | $204$ | $2$ | $2$ | $0$ |
| 204.24.0.bj.1 | $204$ | $2$ | $2$ | $0$ |
| 204.24.1.w.1 | $204$ | $2$ | $2$ | $1$ |
| 204.24.1.x.1 | $204$ | $2$ | $2$ | $1$ |
| 204.24.1.y.1 | $204$ | $2$ | $2$ | $1$ |
| 204.24.1.z.1 | $204$ | $2$ | $2$ | $1$ |
| 204.24.2.a.1 | $204$ | $2$ | $2$ | $2$ |
| 204.24.2.b.1 | $204$ | $2$ | $2$ | $2$ |
| 204.24.2.c.1 | $204$ | $2$ | $2$ | $2$ |
| 204.24.2.d.1 | $204$ | $2$ | $2$ | $2$ |
| 204.216.15.fc.1 | $204$ | $18$ | $18$ | $15$ |
| 228.24.0.be.1 | $228$ | $2$ | $2$ | $0$ |
| 228.24.0.bf.1 | $228$ | $2$ | $2$ | $0$ |
| 228.24.0.bg.1 | $228$ | $2$ | $2$ | $0$ |
| 228.24.0.bh.1 | $228$ | $2$ | $2$ | $0$ |
| 228.24.1.w.1 | $228$ | $2$ | $2$ | $1$ |
| 228.24.1.x.1 | $228$ | $2$ | $2$ | $1$ |
| 228.24.1.y.1 | $228$ | $2$ | $2$ | $1$ |
| 228.24.1.z.1 | $228$ | $2$ | $2$ | $1$ |
| 228.24.2.a.1 | $228$ | $2$ | $2$ | $2$ |
| 228.24.2.b.1 | $228$ | $2$ | $2$ | $2$ |
| 228.24.2.c.1 | $228$ | $2$ | $2$ | $2$ |
| 228.24.2.d.1 | $228$ | $2$ | $2$ | $2$ |
| 228.240.20.i.1 | $228$ | $20$ | $20$ | $20$ |
| 264.24.0.ok.1 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0.ol.1 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0.om.1 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0.on.1 | $264$ | $2$ | $2$ | $0$ |
| 264.24.1.oa.1 | $264$ | $2$ | $2$ | $1$ |
| 264.24.1.ob.1 | $264$ | $2$ | $2$ | $1$ |
| 264.24.1.oc.1 | $264$ | $2$ | $2$ | $1$ |
| 264.24.1.od.1 | $264$ | $2$ | $2$ | $1$ |
| 264.24.2.e.1 | $264$ | $2$ | $2$ | $2$ |
| 264.24.2.f.1 | $264$ | $2$ | $2$ | $2$ |
| 264.24.2.g.1 | $264$ | $2$ | $2$ | $2$ |
| 264.24.2.h.1 | $264$ | $2$ | $2$ | $2$ |
| 276.24.0.be.1 | $276$ | $2$ | $2$ | $0$ |
| 276.24.0.bf.1 | $276$ | $2$ | $2$ | $0$ |
| 276.24.0.bg.1 | $276$ | $2$ | $2$ | $0$ |
| 276.24.0.bh.1 | $276$ | $2$ | $2$ | $0$ |
| 276.24.1.w.1 | $276$ | $2$ | $2$ | $1$ |
| 276.24.1.x.1 | $276$ | $2$ | $2$ | $1$ |
| 276.24.1.y.1 | $276$ | $2$ | $2$ | $1$ |
| 276.24.1.z.1 | $276$ | $2$ | $2$ | $1$ |
| 276.24.2.a.1 | $276$ | $2$ | $2$ | $2$ |
| 276.24.2.b.1 | $276$ | $2$ | $2$ | $2$ |
| 276.24.2.c.1 | $276$ | $2$ | $2$ | $2$ |
| 276.24.2.d.1 | $276$ | $2$ | $2$ | $2$ |
| 276.288.24.i.1 | $276$ | $24$ | $24$ | $24$ |
| 312.24.0.oq.1 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0.or.1 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0.os.1 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0.ot.1 | $312$ | $2$ | $2$ | $0$ |
| 312.24.1.oa.1 | $312$ | $2$ | $2$ | $1$ |
| 312.24.1.ob.1 | $312$ | $2$ | $2$ | $1$ |
| 312.24.1.oc.1 | $312$ | $2$ | $2$ | $1$ |
| 312.24.1.od.1 | $312$ | $2$ | $2$ | $1$ |
| 312.24.2.a.1 | $312$ | $2$ | $2$ | $2$ |
| 312.24.2.b.1 | $312$ | $2$ | $2$ | $2$ |
| 312.24.2.c.1 | $312$ | $2$ | $2$ | $2$ |
| 312.24.2.d.1 | $312$ | $2$ | $2$ | $2$ |