Invariants
Level: | $60$ | $\SL_2$-level: | $20$ | Newform level: | $900$ | ||
Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
Cusps: | $12$ (none of which are rational) | Cusp widths | $1^{4}\cdot4^{2}\cdot5^{4}\cdot20^{2}$ | Cusp orbits | $2^{2}\cdot4^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $1$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 20H1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.72.1.331 |
Level structure
$\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}1&10\\0&17\end{bmatrix}$, $\begin{bmatrix}9&5\\34&27\end{bmatrix}$, $\begin{bmatrix}13&15\\38&17\end{bmatrix}$, $\begin{bmatrix}51&55\\40&7\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 60-isogeny field degree: | $8$ |
Cyclic 60-torsion field degree: | $128$ |
Full 60-torsion field degree: | $30720$ |
Jacobian
Conductor: | $2^{2}\cdot3^{2}\cdot5^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 900.2.a.b |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 3 y z - w^{2} $ |
$=$ | $5 x^{2} + y^{2} + y z + 5 z^{2} - w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 45 x^{4} + 5 x^{2} y^{2} - 6 x^{2} z^{2} + z^{4} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ | $=$ | $\displaystyle z$ |
$\displaystyle Y$ | $=$ | $\displaystyle 3x$ |
$\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 3\,\frac{2187y^{18}-21870y^{16}w^{2}+98415y^{14}w^{4}-265680y^{12}w^{6}+484380y^{10}w^{8}-636120y^{8}w^{10}+625020y^{6}w^{12}-469200y^{4}w^{14}+271450y^{2}w^{16}+273375z^{18}-546750z^{16}w^{2}+1366875z^{14}w^{4}-1782000z^{12}w^{6}+2227500z^{10}w^{8}-1971000z^{8}w^{10}+1469100z^{6}w^{12}-850800z^{4}w^{14}+369650z^{2}w^{16}-120900w^{18}}{w^{4}z^{2}(3z^{2}-w^{2})^{5}(15z^{2}-w^{2})}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
20.36.0.a.1 | $20$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
60.36.0.d.1 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
60.36.1.bh.1 | $60$ | $2$ | $2$ | $1$ | $1$ | 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 |
---|---|---|---|---|---|---|
60.144.5.bn.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
60.144.5.dh.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
60.144.5.ki.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
60.144.5.ko.1 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
60.144.5.ol.2 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
60.144.5.oz.2 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
60.144.5.pz.2 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
60.144.5.qf.2 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
60.216.13.gs.1 | $60$ | $3$ | $3$ | $13$ | $2$ | $1^{6}\cdot2^{3}$ |
60.288.13.nw.1 | $60$ | $4$ | $4$ | $13$ | $2$ | $1^{6}\cdot2^{3}$ |
60.360.13.bt.1 | $60$ | $5$ | $5$ | $13$ | $2$ | $1^{6}\cdot2^{3}$ |
120.144.5.it.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.xa.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.dbi.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.dcy.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.ecr.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.egl.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.enl.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.epb.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
300.360.13.bi.1 | $300$ | $5$ | $5$ | $13$ | $?$ | not computed |