Invariants
| Level: | $60$ | $\SL_2$-level: | $12$ | Newform level: | $3600$ | ||
| Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $4 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $12^{6}$ | Cusp orbits | $2\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $3$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12A4 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.72.4.74 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}19&40\\28&11\end{bmatrix}$, $\begin{bmatrix}23&16\\59&17\end{bmatrix}$, $\begin{bmatrix}27&20\\17&33\end{bmatrix}$, $\begin{bmatrix}33&34\\7&3\end{bmatrix}$ |
| 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: | $30720$ |
Jacobian
| Conductor: | $2^{10}\cdot3^{8}\cdot5^{4}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}$ |
| Newforms: | 36.2.a.a, 144.2.a.a, 900.2.a.g$^{2}$ |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 5 x^{2} - 2 z^{2} - 2 z w + w^{2} $ |
| $=$ | $x z^{2} + x z w + x w^{2} + 20 y^{3}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{6} - 60 x^{4} z^{2} - 4 x^{3} y^{3} + 900 x^{2} z^{4} + 120 x y^{3} z^{2} + y^{6} $ |
Rational points
This modular curve has no $\Q_p$ points for $p=7$, and therefore no rational points.
Maps between models of this curve
Birational map from canonical model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{5}z$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^8\cdot3^3\,\frac{z^{3}(z-w)^{3}(z+w)^{3}(z+2w)^{3}}{(z^{2}+zw+w^{2})^{4}(2z^{2}+2zw-w^{2})^{2}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.36.2.bw.1 | $12$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
| 60.24.0.l.1 | $60$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 60.36.1.fr.1 | $60$ | $2$ | $2$ | $1$ | $0$ | $1^{3}$ |
| 60.36.1.ft.1 | $60$ | $2$ | $2$ | $1$ | $0$ | $1^{3}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.144.7.ip.1 | $60$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 60.144.7.ir.1 | $60$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 60.144.7.jd.1 | $60$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 60.144.7.jh.1 | $60$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 60.144.7.kw.1 | $60$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 60.144.7.la.1 | $60$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 60.144.7.ll.1 | $60$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 60.144.7.ln.1 | $60$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 60.360.28.cy.1 | $60$ | $5$ | $5$ | $28$ | $11$ | $1^{24}$ |
| 60.432.31.gu.1 | $60$ | $6$ | $6$ | $31$ | $7$ | $1^{27}$ |
| 60.720.55.xo.1 | $60$ | $10$ | $10$ | $55$ | $19$ | $1^{51}$ |
| 120.144.7.gcf.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.144.7.gdj.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.144.7.gjr.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.144.7.glj.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.144.7.gzp.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.144.7.hbh.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.144.7.hhk.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.144.7.hiq.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.144.9.efx.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.egh.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.ehv.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.eib.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.jvp.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.jvv.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.jwh.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.jwj.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.lev.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.lfb.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.ljv.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.ljx.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.lzt.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.lzz.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.mcx.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.mcz.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.ohv.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.ohx.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.okv.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.olb.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.pkd.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.pkf.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.poz.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.ppf.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.qen.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.qep.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.qfb.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.qfh.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.qvt.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.qvz.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.qxn.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.qxx.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 180.216.16.bz.1 | $180$ | $3$ | $3$ | $16$ | $?$ | not computed |