Invariants
| Level: | $60$ | $\SL_2$-level: | $12$ | Newform level: | $3600$ | ||
| Index: | $18$ | $\PSL_2$-index: | $18$ | ||||
| Genus: | $1 = 1 + \frac{ 18 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}$ | ||||||
| Cusps: | $3$ (of which $1$ is rational) | Cusp widths | $3^{2}\cdot12$ | Cusp orbits | $1\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $1$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 12B1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.18.1.6 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}3&2\\14&15\end{bmatrix}$, $\begin{bmatrix}5&14\\38&1\end{bmatrix}$, $\begin{bmatrix}19&36\\3&49\end{bmatrix}$, $\begin{bmatrix}31&38\\49&31\end{bmatrix}$, $\begin{bmatrix}51&26\\29&57\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 60-isogeny field degree: | $48$ |
| Cyclic 60-torsion field degree: | $768$ |
| Full 60-torsion field degree: | $122880$ |
Jacobian
| Conductor: | $2^{4}\cdot3^{2}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 3600.2.a.e |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 125 $ |
Rational points
This modular curve has 1 rational cusp and 1 rational CM point, but no other known rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 18 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^6}{5^6}\cdot\frac{(4y^{2}+375z^{2})^{3}}{z^{4}(y^{2}+125z^{2})}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
| 20.6.0.b.1 | $20$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 6.9.0.a.1 | $6$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 20.6.0.b.1 | $20$ | $3$ | $3$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 60.36.1.bl.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.36.1.bm.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.36.1.bz.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.36.1.ca.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.36.1.cn.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.36.1.co.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.36.1.db.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.36.1.dc.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.36.2.a.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.36.2.i.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.36.2.bg.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.36.2.bi.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.36.2.cq.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.36.2.cr.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.36.2.dg.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.36.2.dh.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.36.2.dy.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.36.2.dz.1 | $60$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 60.36.2.eo.1 | $60$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 60.36.2.ep.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.36.2.fe.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.36.2.fg.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.36.2.fj.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.36.2.fk.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.90.7.j.1 | $60$ | $5$ | $5$ | $7$ | $5$ | $1^{6}$ |
| 60.108.7.h.1 | $60$ | $6$ | $6$ | $7$ | $0$ | $1^{6}$ |
| 60.180.13.jz.1 | $60$ | $10$ | $10$ | $13$ | $7$ | $1^{12}$ |
| 120.36.1.em.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.36.1.ep.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.36.1.gi.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.36.1.gl.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.36.1.ie.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.36.1.ih.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.36.1.ka.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.36.1.kd.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.36.2.i.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.z.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.ej.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.ep.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.im.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.ip.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.ki.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.kl.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.mg.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.mj.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.oc.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.of.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.py.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.qe.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.qo.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.qr.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 180.54.4.b.1 | $180$ | $3$ | $3$ | $4$ | $?$ | not computed |
| 180.162.10.h.1 | $180$ | $9$ | $9$ | $10$ | $?$ | not computed |