Invariants
| Level: | $12$ | $\SL_2$-level: | $12$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (all of which are rational) | Cusp widths | $1^{2}\cdot3^{2}\cdot4\cdot12$ | Cusp orbits | $1^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $6$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12E0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.24.0.3 |
Level structure
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 330 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^4}\cdot\frac{x^{24}(3x^{2}-4y^{2})^{3}(3x^{6}-12x^{4}y^{2}+144x^{2}y^{4}-64y^{6})^{3}}{y^{4}x^{36}(x-2y)^{3}(x+2y)^{3}(3x-2y)(3x+2y)}$ |
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 |
|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $6$ | $6$ | $0$ | $0$ |
| $X_0(4)$ | $4$ | $4$ | $4$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_0(4)$ | $4$ | $4$ | $4$ | $0$ | $0$ |
| $X_0(6)$ | $6$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| $X_{\pm1}(12)$ | $12$ | $2$ | $2$ | $0$ |
| 12.48.0.c.2 | $12$ | $2$ | $2$ | $0$ |
| 12.48.0.c.3 | $12$ | $2$ | $2$ | $0$ |
| 12.48.0.c.4 | $12$ | $2$ | $2$ | $0$ |
| 12.48.1.b.1 | $12$ | $2$ | $2$ | $1$ |
| 12.48.1.h.1 | $12$ | $2$ | $2$ | $1$ |
| 12.48.1.k.1 | $12$ | $2$ | $2$ | $1$ |
| 12.48.1.l.1 | $12$ | $2$ | $2$ | $1$ |
| 12.72.1.f.1 | $12$ | $3$ | $3$ | $1$ |
| 24.48.0.bs.1 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0.bs.2 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0.bt.1 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0.bt.2 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0.bu.1 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0.bu.2 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0.bu.3 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0.bu.4 | $24$ | $2$ | $2$ | $0$ |
| 24.48.1.cg.1 | $24$ | $2$ | $2$ | $1$ |
| 24.48.1.es.1 | $24$ | $2$ | $2$ | $1$ |
| 24.48.1.ik.1 | $24$ | $2$ | $2$ | $1$ |
| 24.48.1.in.1 | $24$ | $2$ | $2$ | $1$ |
| 24.48.1.iq.1 | $24$ | $2$ | $2$ | $1$ |
| $X_0(24)$ | $24$ | $2$ | $2$ | $1$ |
| 24.48.1.is.1 | $24$ | $2$ | $2$ | $1$ |
| 24.48.1.it.1 | $24$ | $2$ | $2$ | $1$ |
| 24.48.1.iu.1 | $24$ | $2$ | $2$ | $1$ |
| 24.48.1.iv.1 | $24$ | $2$ | $2$ | $1$ |
| 24.48.1.iw.1 | $24$ | $2$ | $2$ | $1$ |
| 24.48.1.ix.1 | $24$ | $2$ | $2$ | $1$ |
| 24.48.2.f.1 | $24$ | $2$ | $2$ | $2$ |
| 24.48.2.f.2 | $24$ | $2$ | $2$ | $2$ |
| 24.48.2.g.1 | $24$ | $2$ | $2$ | $2$ |
| 24.48.2.g.2 | $24$ | $2$ | $2$ | $2$ |
| $X_0(36)$ | $36$ | $3$ | $3$ | $1$ |
| 36.72.4.d.1 | $36$ | $3$ | $3$ | $4$ |
| 36.72.4.f.1 | $36$ | $3$ | $3$ | $4$ |
| 60.48.0.c.1 | $60$ | $2$ | $2$ | $0$ |
| 60.48.0.c.2 | $60$ | $2$ | $2$ | $0$ |
| 60.48.0.c.3 | $60$ | $2$ | $2$ | $0$ |
| 60.48.0.c.4 | $60$ | $2$ | $2$ | $0$ |
| 60.48.1.k.1 | $60$ | $2$ | $2$ | $1$ |
| 60.48.1.l.1 | $60$ | $2$ | $2$ | $1$ |
| 60.48.1.o.1 | $60$ | $2$ | $2$ | $1$ |
| 60.48.1.p.1 | $60$ | $2$ | $2$ | $1$ |
| 60.120.8.o.1 | $60$ | $5$ | $5$ | $8$ |
| $X_0(60)$ | $60$ | $6$ | $6$ | $7$ |
| 60.240.15.bq.1 | $60$ | $10$ | $10$ | $15$ |
| 84.48.0.c.1 | $84$ | $2$ | $2$ | $0$ |
| 84.48.0.c.2 | $84$ | $2$ | $2$ | $0$ |
| 84.48.0.c.3 | $84$ | $2$ | $2$ | $0$ |
| 84.48.0.c.4 | $84$ | $2$ | $2$ | $0$ |
| 84.48.1.k.1 | $84$ | $2$ | $2$ | $1$ |
| 84.48.1.l.1 | $84$ | $2$ | $2$ | $1$ |
| 84.48.1.o.1 | $84$ | $2$ | $2$ | $1$ |
| 84.48.1.p.1 | $84$ | $2$ | $2$ | $1$ |
| $X_0(84)$ | $84$ | $8$ | $8$ | $11$ |
| 120.48.0.dq.1 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0.dq.2 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0.dr.1 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0.dr.2 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0.ds.1 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0.ds.2 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0.ds.3 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0.ds.4 | $120$ | $2$ | $2$ | $0$ |
| 120.48.1.zc.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1.zf.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1.zo.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1.zr.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1.zu.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1.zv.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1.zw.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1.zx.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1.zy.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1.zz.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1.baa.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1.bab.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.2.h.1 | $120$ | $2$ | $2$ | $2$ |
| 120.48.2.h.2 | $120$ | $2$ | $2$ | $2$ |
| 120.48.2.i.1 | $120$ | $2$ | $2$ | $2$ |
| 120.48.2.i.2 | $120$ | $2$ | $2$ | $2$ |
| 132.48.0.c.1 | $132$ | $2$ | $2$ | $0$ |
| 132.48.0.c.2 | $132$ | $2$ | $2$ | $0$ |
| 132.48.0.c.3 | $132$ | $2$ | $2$ | $0$ |
| 132.48.0.c.4 | $132$ | $2$ | $2$ | $0$ |
| 132.48.1.k.1 | $132$ | $2$ | $2$ | $1$ |
| 132.48.1.l.1 | $132$ | $2$ | $2$ | $1$ |
| 132.48.1.o.1 | $132$ | $2$ | $2$ | $1$ |
| 132.48.1.p.1 | $132$ | $2$ | $2$ | $1$ |
| $X_0(132)$ | $132$ | $12$ | $12$ | $19$ |
| 156.48.0.c.1 | $156$ | $2$ | $2$ | $0$ |
| 156.48.0.c.2 | $156$ | $2$ | $2$ | $0$ |
| 156.48.0.c.3 | $156$ | $2$ | $2$ | $0$ |
| 156.48.0.c.4 | $156$ | $2$ | $2$ | $0$ |
| 156.48.1.k.1 | $156$ | $2$ | $2$ | $1$ |
| 156.48.1.l.1 | $156$ | $2$ | $2$ | $1$ |
| 156.48.1.o.1 | $156$ | $2$ | $2$ | $1$ |
| 156.48.1.p.1 | $156$ | $2$ | $2$ | $1$ |
| $X_0(156)$ | $156$ | $14$ | $14$ | $23$ |
| 168.48.0.do.1 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0.do.2 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0.dp.1 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0.dp.2 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0.dq.1 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0.dq.2 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0.dq.3 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0.dq.4 | $168$ | $2$ | $2$ | $0$ |
| 168.48.1.za.1 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1.zd.1 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1.zm.1 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1.zp.1 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1.zs.1 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1.zt.1 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1.zu.1 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1.zv.1 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1.zw.1 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1.zx.1 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1.zy.1 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1.zz.1 | $168$ | $2$ | $2$ | $1$ |
| 168.48.2.f.1 | $168$ | $2$ | $2$ | $2$ |
| 168.48.2.f.2 | $168$ | $2$ | $2$ | $2$ |
| 168.48.2.g.1 | $168$ | $2$ | $2$ | $2$ |
| 168.48.2.g.2 | $168$ | $2$ | $2$ | $2$ |
| 204.48.0.c.1 | $204$ | $2$ | $2$ | $0$ |
| 204.48.0.c.2 | $204$ | $2$ | $2$ | $0$ |
| 204.48.0.c.3 | $204$ | $2$ | $2$ | $0$ |
| 204.48.0.c.4 | $204$ | $2$ | $2$ | $0$ |
| 204.48.1.k.1 | $204$ | $2$ | $2$ | $1$ |
| 204.48.1.l.1 | $204$ | $2$ | $2$ | $1$ |
| 204.48.1.o.1 | $204$ | $2$ | $2$ | $1$ |
| 204.48.1.p.1 | $204$ | $2$ | $2$ | $1$ |
| 228.48.0.c.1 | $228$ | $2$ | $2$ | $0$ |
| 228.48.0.c.2 | $228$ | $2$ | $2$ | $0$ |
| 228.48.0.c.3 | $228$ | $2$ | $2$ | $0$ |
| 228.48.0.c.4 | $228$ | $2$ | $2$ | $0$ |
| 228.48.1.k.1 | $228$ | $2$ | $2$ | $1$ |
| 228.48.1.l.1 | $228$ | $2$ | $2$ | $1$ |
| 228.48.1.o.1 | $228$ | $2$ | $2$ | $1$ |
| 228.48.1.p.1 | $228$ | $2$ | $2$ | $1$ |
| 264.48.0.do.1 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0.do.2 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0.dp.1 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0.dp.2 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0.dq.1 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0.dq.2 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0.dq.3 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0.dq.4 | $264$ | $2$ | $2$ | $0$ |
| 264.48.1.za.1 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1.zd.1 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1.zm.1 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1.zp.1 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1.zs.1 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1.zt.1 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1.zu.1 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1.zv.1 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1.zw.1 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1.zx.1 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1.zy.1 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1.zz.1 | $264$ | $2$ | $2$ | $1$ |
| 264.48.2.f.1 | $264$ | $2$ | $2$ | $2$ |
| 264.48.2.f.2 | $264$ | $2$ | $2$ | $2$ |
| 264.48.2.g.1 | $264$ | $2$ | $2$ | $2$ |
| 264.48.2.g.2 | $264$ | $2$ | $2$ | $2$ |
| 276.48.0.c.1 | $276$ | $2$ | $2$ | $0$ |
| 276.48.0.c.2 | $276$ | $2$ | $2$ | $0$ |
| 276.48.0.c.3 | $276$ | $2$ | $2$ | $0$ |
| 276.48.0.c.4 | $276$ | $2$ | $2$ | $0$ |
| 276.48.1.k.1 | $276$ | $2$ | $2$ | $1$ |
| 276.48.1.l.1 | $276$ | $2$ | $2$ | $1$ |
| 276.48.1.o.1 | $276$ | $2$ | $2$ | $1$ |
| 276.48.1.p.1 | $276$ | $2$ | $2$ | $1$ |
| 312.48.0.dq.1 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0.dq.2 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0.dr.1 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0.dr.2 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0.ds.1 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0.ds.2 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0.ds.3 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0.ds.4 | $312$ | $2$ | $2$ | $0$ |
| 312.48.1.zc.1 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1.zf.1 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1.zo.1 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1.zr.1 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1.zu.1 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1.zv.1 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1.zw.1 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1.zx.1 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1.zy.1 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1.zz.1 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1.baa.1 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1.bab.1 | $312$ | $2$ | $2$ | $1$ |
| 312.48.2.h.1 | $312$ | $2$ | $2$ | $2$ |
| 312.48.2.h.2 | $312$ | $2$ | $2$ | $2$ |
| 312.48.2.i.1 | $312$ | $2$ | $2$ | $2$ |
| 312.48.2.i.2 | $312$ | $2$ | $2$ | $2$ |