Invariants
| Level: | $60$ | $\SL_2$-level: | $12$ | Newform level: | $36$ | ||
| Index: | $72$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $1 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (all of which are rational) | Cusp widths | $3^{4}\cdot12^{2}$ | Cusp orbits | $1^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $6$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12K1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.72.1.136 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}15&4\\29&3\end{bmatrix}$, $\begin{bmatrix}33&16\\28&15\end{bmatrix}$, $\begin{bmatrix}41&36\\27&35\end{bmatrix}$, $\begin{bmatrix}59&0\\3&19\end{bmatrix}$, $\begin{bmatrix}59&36\\21&53\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.36.1.m.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $12$ |
| Cyclic 60-torsion field degree: | $192$ |
| Full 60-torsion field degree: | $30720$ |
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} - 15x + 22 $ |
Rational points
This modular curve has 6 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(2:0:1)$, $(-1:-6:1)$, $(0:1:0)$, $(3:2:1)$, $(3:-2:1)$, $(-1:6:1)$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{18x^{2}y^{10}-12582x^{2}y^{8}z^{2}+2119608x^{2}y^{6}z^{4}-144324018x^{2}y^{4}z^{6}+4356546282x^{2}y^{2}z^{8}-48659601825x^{2}z^{10}-207xy^{10}z+88344xy^{8}z^{3}-12119949xy^{6}z^{5}+734300316xy^{4}z^{7}-20534618529xy^{2}z^{9}+217221364926xz^{11}-y^{12}+1674y^{10}z^{2}-397170y^{8}z^{4}+36023562y^{6}z^{6}-1539259443y^{4}z^{8}+31170743838y^{2}z^{10}-239804302869z^{12}}{z^{4}(18x^{2}y^{6}-9648x^{2}y^{4}z^{2}+1004832x^{2}y^{2}z^{4}-28159488x^{2}z^{6}-195xy^{6}z+61308xy^{4}z^{3}-5114448xy^{2}z^{5}+125706816xz^{7}-y^{8}+1426y^{6}z^{2}-224424y^{4}z^{4}+10565856y^{2}z^{6}-138775680z^{8})}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.144.1-12.f.1.4 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.144.1-12.f.1.7 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.144.1-12.g.1.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.144.1-12.g.1.4 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.144.1-60.i.1.3 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.144.1-60.i.1.4 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.144.1-60.j.1.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.144.1-60.j.1.4 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.144.3-12.k.1.7 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-12.bf.1.4 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-12.bx.1.2 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-12.by.1.2 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-12.cc.1.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-12.cc.1.2 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-12.cd.1.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-12.cd.1.4 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-60.ga.1.2 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-60.gc.1.2 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.144.3-60.gi.1.2 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.144.3-60.gk.1.2 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-60.hy.1.4 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.144.3-60.hy.1.7 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.144.3-60.hz.1.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.144.3-60.hz.1.2 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.360.13-60.t.1.10 | $60$ | $5$ | $5$ | $13$ | $3$ | $1^{12}$ |
| 60.432.13-60.bm.1.14 | $60$ | $6$ | $6$ | $13$ | $0$ | $1^{12}$ |
| 60.720.25-60.cx.1.14 | $60$ | $10$ | $10$ | $25$ | $5$ | $1^{24}$ |
| 120.144.1-24.s.1.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.144.1-24.s.1.10 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.144.1-24.v.1.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.144.1-24.v.1.9 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.144.1-120.be.1.11 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.144.1-120.be.1.23 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.144.1-120.bh.1.7 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.144.1-120.bh.1.19 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.144.3-24.ce.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.hw.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.mq.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.mx.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.oq.1.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.oq.1.19 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.or.1.12 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.or.1.27 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.pc.1.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.pc.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.pc.1.24 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.pc.1.26 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.pd.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.pd.1.13 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.pd.1.20 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.pd.1.30 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.pe.1.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.pe.1.15 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.pf.1.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.pf.1.19 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.pi.1.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.pi.1.19 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.pj.1.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.pj.1.15 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.pm.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.pm.1.13 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.pm.1.20 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.pm.1.30 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.pn.1.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.pn.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.pn.1.24 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.pn.1.26 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.po.1.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.po.1.15 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.pp.1.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.pp.1.19 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.qg.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.qg.1.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.qj.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.qj.1.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bmj.1.11 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bmx.1.11 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bon.1.11 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bpb.1.11 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bzw.1.30 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bzw.1.31 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bzx.1.22 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bzx.1.23 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.caa.1.29 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.caa.1.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.caa.1.33 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.caa.1.36 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cab.1.21 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cab.1.24 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cab.1.41 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cab.1.44 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cac.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cac.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cad.1.18 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cad.1.19 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cao.1.18 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cao.1.19 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cap.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cap.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cas.1.17 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cas.1.20 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cas.1.45 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cas.1.48 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cat.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cat.1.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cat.1.61 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cat.1.64 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cau.1.22 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cau.1.23 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cav.1.30 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cav.1.31 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cgw.1.19 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cgw.1.20 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cgz.1.5 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cgz.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 180.216.4-36.c.1.2 | $180$ | $3$ | $3$ | $4$ | $?$ | not computed |
| 180.216.4-36.c.1.8 | $180$ | $3$ | $3$ | $4$ | $?$ | not computed |
| 180.216.7-36.q.1.4 | $180$ | $3$ | $3$ | $7$ | $?$ | not computed |