Invariants
Level: | $60$ | $\SL_2$-level: | $20$ | Newform level: | $3600$ | ||
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: | 60.72.1.333 |
Level structure
$\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}23&25\\10&51\end{bmatrix}$, $\begin{bmatrix}27&20\\26&9\end{bmatrix}$, $\begin{bmatrix}29&10\\6&49\end{bmatrix}$, $\begin{bmatrix}41&40\\30&53\end{bmatrix}$ |
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: | $2^{4}\cdot3^{2}\cdot5^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 3600.2.a.be |
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.c.1 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
60.36.1.bg.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.bm.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
60.144.5.ct.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
60.144.5.kk.1 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
60.144.5.km.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
60.144.5.or.2 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
60.144.5.ot.2 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
60.144.5.qb.2 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
60.144.5.qd.2 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
60.216.13.gq.1 | $60$ | $3$ | $3$ | $13$ | $1$ | $1^{6}\cdot2^{3}$ |
60.288.13.nu.1 | $60$ | $4$ | $4$ | $13$ | $4$ | $1^{6}\cdot2^{3}$ |
60.360.13.bs.1 | $60$ | $5$ | $5$ | $13$ | $3$ | $1^{6}\cdot2^{3}$ |
120.144.5.ir.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.tg.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.dbw.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.dck.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.eeh.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.eev.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.enz.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.eon.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
300.360.13.bg.1 | $300$ | $5$ | $5$ | $13$ | $?$ | not computed |