Invariants
Level: | $60$ | $\SL_2$-level: | $30$ | Newform level: | $900$ | ||
Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 16 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (none of which are rational) | Cusp widths | $6^{2}\cdot30^{2}$ | Cusp orbits | $2^{2}$ | ||
Elliptic points: | $16$ 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: | 30D1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.72.1.426 |
Level structure
$\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}17&35\\53&46\end{bmatrix}$, $\begin{bmatrix}26&15\\45&7\end{bmatrix}$, $\begin{bmatrix}31&20\\31&7\end{bmatrix}$, $\begin{bmatrix}44&55\\25&13\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 60-isogeny field degree: | $24$ |
Cyclic 60-torsion field degree: | $384$ |
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 $ | $=$ | $ 5 x^{2} - 2 x y + y^{2} + z^{2} $ |
$=$ | $5 x^{2} + 13 x y + y^{2} + z^{2} - w^{2}$ |
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:
$\displaystyle X$ | $=$ | $\displaystyle y$ |
$\displaystyle Y$ | $=$ | $\displaystyle 3z$ |
$\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 3^3\,\frac{22245890625000y^{2}z^{16}-23728950000000y^{2}z^{14}w^{2}+7613038125000y^{2}z^{12}w^{4}-381996000000y^{2}z^{10}w^{6}-134227125000y^{2}z^{8}w^{8}+6120360000y^{2}z^{6}w^{10}+480951000y^{2}z^{4}w^{12}+4219200y^{2}z^{2}w^{14}-21120y^{2}w^{16}-1423828125z^{18}+1708593750z^{16}w^{2}+493726640625z^{14}w^{4}-461062125000z^{12}w^{6}+118422253125z^{10}w^{8}-1419221250z^{8}w^{10}-1485887625z^{6}w^{12}-18773100z^{4}w^{14}+783120z^{2}w^{16}+8896w^{18}}{w^{6}(11390625y^{2}z^{10}-7593750y^{2}z^{8}w^{2}+1012500y^{2}z^{6}w^{4}+135000y^{2}z^{4}w^{6}-19125y^{2}z^{2}w^{8}-330y^{2}w^{10}+11390625z^{12}-9112500z^{10}w^{2}+1771875z^{8}w^{4}+135000z^{6}w^{6}-47250z^{4}w^{8}+120z^{2}w^{10}+139w^{12})}$ |
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 |
---|---|---|---|---|---|---|
30.36.1.p.1 | $30$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
60.36.0.i.1 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
60.36.0.cg.2 | $60$ | $2$ | $2$ | $0$ | $0$ | 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 |
---|---|---|---|---|---|---|
60.144.9.bq.2 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
60.144.9.bt.2 | $60$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
60.144.9.dx.1 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
60.144.9.dz.1 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
60.144.9.gf.1 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
60.144.9.gj.1 | $60$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
60.144.9.hu.2 | $60$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
60.144.9.hy.2 | $60$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
60.216.9.u.2 | $60$ | $3$ | $3$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
60.288.13.sd.2 | $60$ | $4$ | $4$ | $13$ | $5$ | $1^{6}\cdot2^{3}$ |
60.360.21.co.1 | $60$ | $5$ | $5$ | $21$ | $6$ | $1^{8}\cdot2^{6}$ |
120.144.9.jbo.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.jcj.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.npv.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.nqj.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.rxz.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.rzb.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.sjr.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.skt.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
180.216.13.ih.1 | $180$ | $3$ | $3$ | $13$ | $?$ | not computed |
300.360.21.n.2 | $300$ | $5$ | $5$ | $21$ | $?$ | not computed |