Invariants
| Level: | $60$ | $\SL_2$-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: | 60.48.0.59 |
Level structure
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 12 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^4\cdot5^3}\cdot\frac{x^{48}(75x^{4}-480x^{2}y^{2}-256y^{4})^{3}(46875x^{12}-11700000x^{10}y^{2}+39840000x^{8}y^{4}-30720000x^{6}y^{6}-44236800x^{4}y^{8}+31457280x^{2}y^{10}-16777216y^{12})^{3}}{y^{2}x^{54}(5x^{2}-16y^{2})^{3}(5x^{2}+16y^{2})^{12}(15x^{2}-16y^{2})^{4}(15x^{2}+16y^{2})}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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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 |
|---|---|---|---|---|
| 60.96.1.b.1 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1.i.2 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1.j.1 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1.k.2 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1.l.1 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1.m.4 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1.n.1 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1.o.4 | $60$ | $2$ | $2$ | $1$ |
| 60.144.3.g.1 | $60$ | $3$ | $3$ | $3$ |
| 60.240.16.f.1 | $60$ | $5$ | $5$ | $16$ |
| 60.288.15.h.2 | $60$ | $6$ | $6$ | $15$ |
| 60.480.31.j.3 | $60$ | $10$ | $10$ | $31$ |
| 120.96.1.qh.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.qu.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.qy.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.ra.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.rb.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.re.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.rf.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.ri.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.rk.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.rl.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.ro.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.rp.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.rt.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.rw.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.rz.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.sc.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.sd.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.sg.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.sh.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.sk.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.tc.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.td.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.tg.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.th.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.3.sb.3 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.sc.3 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.sf.4 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.sg.4 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.sy.4 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.tb.4 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.tc.3 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.tf.3 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.th.3 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.ti.3 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.tl.4 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.tm.4 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.to.4 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.tr.4 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.ts.3 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.tv.3 | $120$ | $2$ | $2$ | $3$ |
| 180.144.3.c.2 | $180$ | $3$ | $3$ | $3$ |
| 180.144.8.e.2 | $180$ | $3$ | $3$ | $8$ |
| 180.144.8.f.2 | $180$ | $3$ | $3$ | $8$ |