Invariants
| Level: | $60$ | $\SL_2$-level: | $12$ | Newform level: | $36$ | ||
| Index: | $36$ | $\PSL_2$-index: | $18$ | ||||
| Genus: | $1 = 1 + \frac{ 18 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}$ | ||||||
| Cusps: | $3$ (all of which are rational) | Cusp widths | $3^{2}\cdot12$ | Cusp orbits | $1^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $3$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4,-16$) |
Other labels
| Cummins and Pauli (CP) label: | 12B1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.36.1.131 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}11&24\\15&1\end{bmatrix}$, $\begin{bmatrix}19&24\\42&53\end{bmatrix}$, $\begin{bmatrix}43&36\\51&1\end{bmatrix}$, $\begin{bmatrix}47&36\\24&11\end{bmatrix}$, $\begin{bmatrix}55&56\\23&59\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.18.1.c.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $24$ |
| Cyclic 60-torsion field degree: | $384$ |
| Full 60-torsion field degree: | $61440$ |
Jacobian
| Conductor: | $2^{2}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 36.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + 1 $ |
Rational points
This modular curve has 3 rational cusps and 2 rational CM points, but no other known rational points. The following are the known rational points on this modular curve (one row per $j$-invariant).
| Elliptic curve | CM | $j$-invariant | $j$-height | Weierstrass model | |
|---|---|---|---|---|---|
| no | $\infty$ | $0.000$ | $(0:1:0)$, $(0:1:1)$, $(0:-1:1)$ | ||
| 32.a3 | $-4$ | $1728$ | $= 2^{6} \cdot 3^{3}$ | $7.455$ | $(-1:0:1)$ |
| 32.a1 | $-16$ | $287496$ | $= 2^{3} \cdot 3^{3} \cdot 11^{3}$ | $12.569$ | $(2:-3:1)$, $(2:3:1)$ |
Maps to other modular curves
$j$-invariant map of degree 18 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{(4y^{2}-3z^{2})^{3}}{z^{4}(y-z)(y+z)}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.12.0-4.c.1.1 | $60$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.72.1-12.i.1.4 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.72.1-60.i.1.3 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.72.1-12.j.1.3 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.72.1-60.j.1.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.72.1-12.m.1.4 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.72.1-60.m.1.4 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.72.1-12.n.1.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.72.1-60.n.1.2 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.72.2-12.b.1.7 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.72.2-12.p.1.5 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.72.2-12.s.1.2 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.72.2-60.s.1.3 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.72.2-12.t.1.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.72.2-60.t.1.3 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.72.2-12.w.1.3 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.72.2-60.w.1.3 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.72.2-12.x.1.2 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.72.2-60.x.1.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.72.2-12.ba.1.2 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.72.2-60.ba.1.8 | $60$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 60.72.2-12.bb.1.3 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.72.2-60.bb.1.8 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.72.2-60.be.1.7 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.72.2-60.bf.1.8 | $60$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 60.180.7-60.c.1.16 | $60$ | $5$ | $5$ | $7$ | $3$ | $1^{6}$ |
| 60.216.7-60.c.1.27 | $60$ | $6$ | $6$ | $7$ | $0$ | $1^{6}$ |
| 60.360.13-60.be.1.23 | $60$ | $10$ | $10$ | $13$ | $5$ | $1^{12}$ |
| 120.72.1-24.be.1.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1-120.be.1.16 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1-24.bh.1.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1-120.bh.1.16 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1-24.bq.1.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1-120.bq.1.14 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1-24.bt.1.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1-120.bt.1.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.2-24.j.1.6 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.bu.1.6 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cc.1.6 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.cc.1.16 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cf.1.6 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.cf.1.16 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.ci.1.6 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.ci.1.27 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cj.1.10 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cj.1.47 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.ck.1.2 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.ck.1.31 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cl.1.6 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cl.1.27 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cm.1.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cm.1.32 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cn.1.5 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cn.1.28 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.co.1.5 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.co.1.28 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.co.1.16 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cp.1.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cp.1.32 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cq.1.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cq.1.32 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cr.1.5 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cr.1.28 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.cr.1.12 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cs.1.5 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cs.1.28 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.ct.1.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.ct.1.32 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cu.1.6 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cu.1.27 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.cu.1.2 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.cu.1.63 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cv.1.2 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cv.1.31 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.cv.1.18 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.cv.1.47 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cw.1.2 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cw.1.31 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.cw.1.7 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.cw.1.58 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cx.1.6 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cx.1.27 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.cx.1.26 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.cx.1.39 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.cy.1.8 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.cy.1.57 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.cz.1.25 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.cz.1.40 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.da.1.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.da.1.64 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.db.1.17 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.db.1.48 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.dc.1.17 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.dc.1.48 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.dd.1.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.dd.1.64 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.de.1.4 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.de.1.25 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.de.1.40 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.df.1.8 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.df.1.57 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.dg.1.26 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.dg.1.39 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.dh.1.6 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.dh.1.7 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.dh.1.58 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.di.1.18 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.di.1.47 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.dj.1.2 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.dj.1.63 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.dq.1.8 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.dq.1.21 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.dt.1.8 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.dt.1.20 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.ec.1.13 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.ef.1.5 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 180.108.4-36.c.1.6 | $180$ | $3$ | $3$ | $4$ | $?$ | not computed |
| 180.324.10-36.f.1.4 | $180$ | $9$ | $9$ | $10$ | $?$ | not computed |