Invariants
| Level: | $60$ | $\SL_2$-level: | $12$ | Newform level: | $144$ | ||
| Index: | $72$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $1 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $6^{6}$ | Cusp orbits | $2\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 6E1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.72.1.167 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}9&38\\59&9\end{bmatrix}$, $\begin{bmatrix}19&12\\24&19\end{bmatrix}$, $\begin{bmatrix}41&22\\50&53\end{bmatrix}$, $\begin{bmatrix}59&0\\39&37\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.36.1.f.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $48$ |
| Cyclic 60-torsion field degree: | $768$ |
| Full 60-torsion field degree: | $30720$ |
Jacobian
| Conductor: | $2^{4}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 144.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 15x - 22 $ |
Rational points
This modular curve has 1 rational CM point but no rational cusps or 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 | |
|---|---|---|---|---|---|
| 32.a3 | $-4$ | $1728$ | $= 2^{6} \cdot 3^{3}$ | $7.455$ | $(-2:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^6\cdot3^3\,\frac{54x^{2}y^{10}-3942x^{2}y^{8}z^{2}+95571576x^{2}y^{6}z^{4}+31818515742x^{2}y^{4}z^{6}+3000412331550x^{2}y^{2}z^{8}+84083780609631x^{2}z^{10}-639xy^{10}z+513648xy^{8}z^{3}+836067987xy^{6}z^{5}+196501877100xy^{4}z^{7}+15271659785295xy^{2}z^{9}+375358541280066xz^{11}-y^{12}-4194y^{10}z^{2}+8961246y^{8}z^{4}+5063882670y^{6}z^{6}+684962338365y^{4}z^{8}+31549476328074y^{2}z^{10}+414381960141291z^{12}}{54x^{2}y^{10}+160758x^{2}y^{8}z^{2}+17993016x^{2}y^{6}z^{4}+42282x^{2}y^{4}z^{6}+21870x^{2}y^{2}z^{8}+6561x^{2}z^{10}+1287xy^{10}z+1126116xy^{8}z^{3}+80446365xy^{6}z^{5}-93312xy^{4}z^{7}-45927xy^{2}z^{9}-13122xz^{11}+y^{12}+17478y^{10}z^{2}+4394682y^{8}z^{4}+88818606y^{6}z^{6}-429381y^{4}z^{8}-231822y^{2}z^{10}-72171z^{12}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.24.1-12.c.1.1 | $60$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 60.36.1-12.a.1.5 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.36.1-12.a.1.6 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.144.3-12.v.1.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-12.ba.1.2 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-12.bh.1.2 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-12.bj.1.2 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-60.dc.1.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-60.de.1.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.144.3-60.eb.1.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-60.ed.1.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.360.13-60.l.1.7 | $60$ | $5$ | $5$ | $13$ | $6$ | $1^{12}$ |
| 60.432.13-60.v.1.11 | $60$ | $6$ | $6$ | $13$ | $3$ | $1^{12}$ |
| 60.720.25-60.bm.1.13 | $60$ | $10$ | $10$ | $25$ | $11$ | $1^{24}$ |
| 120.144.3-24.fl.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.gq.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.ii.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.iw.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.ty.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.um.1.5 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.zw.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bak.1.5 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 180.216.7-36.d.1.3 | $180$ | $3$ | $3$ | $7$ | $?$ | not computed |