Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (of which $2$ are rational) | Cusp widths | $1^{2}\cdot2\cdot3^{2}\cdot4^{2}\cdot6\cdot12^{2}$ | Cusp orbits | $1^{2}\cdot2^{4}$ | ||
| Elliptic points: | $0$ 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: | 12J0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.48.0.922 |
Level structure
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 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2\cdot3^3}\cdot\frac{x^{48}(27x^{4}+36x^{2}y^{2}-4y^{4})^{3}(2187x^{12}+113724x^{10}y^{2}+80676x^{8}y^{4}+12960x^{6}y^{6}-3888x^{4}y^{8}-576x^{2}y^{10}-64y^{12})^{3}}{y^{2}x^{54}(3x^{2}-2y^{2})^{12}(3x^{2}+2y^{2})^{3}(9x^{2}-2y^{2})(9x^{2}+2y^{2})^{4}}$ |
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 |
|---|---|---|---|---|---|
| $X_0(12)$ | $12$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 24.96.1.ck.1 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.cr.4 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.cx.2 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.cy.3 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.cz.1 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.da.2 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.db.1 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.dc.1 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.de.4 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.df.4 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.di.3 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.dj.3 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.dl.3 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.do.3 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.dp.4 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.ds.4 | $24$ | $2$ | $2$ | $1$ |
| 24.96.3.gk.2 | $24$ | $2$ | $2$ | $3$ |
| 24.96.3.gn.2 | $24$ | $2$ | $2$ | $3$ |
| 24.96.3.go.1 | $24$ | $2$ | $2$ | $3$ |
| 24.96.3.gr.1 | $24$ | $2$ | $2$ | $3$ |
| 24.96.3.gt.1 | $24$ | $2$ | $2$ | $3$ |
| 24.96.3.gu.1 | $24$ | $2$ | $2$ | $3$ |
| 24.96.3.gx.2 | $24$ | $2$ | $2$ | $3$ |
| 24.96.3.gy.2 | $24$ | $2$ | $2$ | $3$ |
| 24.144.3.c.2 | $24$ | $3$ | $3$ | $3$ |
| 72.144.3.c.3 | $72$ | $3$ | $3$ | $3$ |
| 72.144.8.m.3 | $72$ | $3$ | $3$ | $8$ |
| 72.144.8.n.3 | $72$ | $3$ | $3$ | $8$ |
| 120.96.1.rr.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.rt.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.ru.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.rw.3 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.rx.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.rz.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.sa.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.sc.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.sm.3 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.sn.3 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.sq.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.sr.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.st.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.sw.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.sx.3 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.ta.3 | $120$ | $2$ | $2$ | $1$ |
| 120.96.3.si.2 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.sl.2 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.sm.1 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.sp.1 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.sr.1 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.ss.1 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.sv.2 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.sw.2 | $120$ | $2$ | $2$ | $3$ |
| 120.240.16.fm.2 | $120$ | $5$ | $5$ | $16$ |
| 120.288.15.ekm.2 | $120$ | $6$ | $6$ | $15$ |
| 168.96.1.rp.4 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.rr.4 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.rs.4 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.ru.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.rv.4 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.rx.3 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.ry.3 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.sa.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.sk.4 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.sl.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.so.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.sp.4 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.sr.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.su.4 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.sv.4 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.sy.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.3.pu.2 | $168$ | $2$ | $2$ | $3$ |
| 168.96.3.px.4 | $168$ | $2$ | $2$ | $3$ |
| 168.96.3.py.4 | $168$ | $2$ | $2$ | $3$ |
| 168.96.3.qb.2 | $168$ | $2$ | $2$ | $3$ |
| 168.96.3.qd.4 | $168$ | $2$ | $2$ | $3$ |
| 168.96.3.qe.2 | $168$ | $2$ | $2$ | $3$ |
| 168.96.3.qh.2 | $168$ | $2$ | $2$ | $3$ |
| 168.96.3.qi.4 | $168$ | $2$ | $2$ | $3$ |
| 168.384.23.mg.2 | $168$ | $8$ | $8$ | $23$ |
| 264.96.1.rp.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.rr.4 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.rs.3 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.ru.3 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.rv.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.rx.4 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.ry.3 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.sa.3 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.sk.4 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.sl.4 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.so.4 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.sp.4 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.sr.4 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.su.4 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.sv.4 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.sy.4 | $264$ | $2$ | $2$ | $1$ |
| 264.96.3.pu.3 | $264$ | $2$ | $2$ | $3$ |
| 264.96.3.px.3 | $264$ | $2$ | $2$ | $3$ |
| 264.96.3.py.3 | $264$ | $2$ | $2$ | $3$ |
| 264.96.3.qb.3 | $264$ | $2$ | $2$ | $3$ |
| 264.96.3.qd.3 | $264$ | $2$ | $2$ | $3$ |
| 264.96.3.qe.3 | $264$ | $2$ | $2$ | $3$ |
| 264.96.3.qh.3 | $264$ | $2$ | $2$ | $3$ |
| 264.96.3.qi.3 | $264$ | $2$ | $2$ | $3$ |
| 312.96.1.rr.4 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.rt.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.ru.4 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.rw.4 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.rx.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.rz.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.sa.4 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.sc.3 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.sm.4 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.sn.4 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.sq.4 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.sr.4 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.st.4 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.sw.4 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.sx.4 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.ta.4 | $312$ | $2$ | $2$ | $1$ |
| 312.96.3.si.4 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.sl.4 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.sm.4 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.sp.4 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.sr.4 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.ss.4 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.sv.4 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.sw.4 | $312$ | $2$ | $2$ | $3$ |