Invariants
Level: | $60$ | $\SL_2$-level: | $30$ | Newform level: | $20$ | ||
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: | $0$ | ||||||
$\Q$-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: | 60.72.1.414 |
Level structure
$\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}16&25\\17&46\end{bmatrix}$, $\begin{bmatrix}34&15\\51&59\end{bmatrix}$, $\begin{bmatrix}51&20\\46&57\end{bmatrix}$, $\begin{bmatrix}58&35\\5&2\end{bmatrix}$ |
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: | $2^{2}\cdot5$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 20.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} - y z $ |
$=$ | $x^{2} + 5 y^{2} + y z + z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 5 x^{4} + 2 x^{2} z^{2} + y^{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:
$\displaystyle X$ | $=$ | $\displaystyle x$ |
$\displaystyle Y$ | $=$ | $\displaystyle w$ |
$\displaystyle Z$ | $=$ | $\displaystyle z$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{118162368yz^{17}-2277754560yz^{15}w^{2}-7343014500yz^{13}w^{4}+32664890250yz^{11}w^{6}-10523081250yz^{9}w^{8}+5575331250yz^{7}w^{10}-805218750yz^{5}w^{12}+77343750yz^{3}w^{14}-7031250yzw^{16}-13931136z^{18}-1200905136z^{16}w^{2}+4705581600z^{14}w^{4}+11568316725z^{12}w^{6}+8333404875z^{10}w^{8}-2773237500z^{8}w^{10}+523884375z^{6}w^{12}-140625000z^{4}w^{14}+7734375z^{2}w^{16}-390625w^{18}}{z^{3}(68381yz^{14}+673915yz^{12}w^{2}+2141025yz^{10}w^{4}+2414375yz^{8}w^{6}-415625yz^{6}w^{8}-2559375yz^{4}w^{10}-1203125yz^{2}w^{12}+78125yw^{14}-8062z^{15}+74318z^{13}w^{2}+737130z^{11}w^{4}+2119750z^{9}w^{6}+2663750z^{7}w^{8}+1286250z^{5}w^{10}-131250z^{3}w^{12}-218750zw^{14})}$ |
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.q.1 | $30$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
60.36.0.i.1 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
60.36.0.ch.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.v.2 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
60.144.9.y.2 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
60.144.9.cu.2 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
60.144.9.cy.1 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
60.144.9.iq.1 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
60.144.9.is.1 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
60.144.9.jl.2 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
60.144.9.jn.2 | $60$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
60.216.9.bp.2 | $60$ | $3$ | $3$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
60.288.13.sp.2 | $60$ | $4$ | $4$ | $13$ | $1$ | $1^{6}\cdot2^{3}$ |
60.360.21.de.1 | $60$ | $5$ | $5$ | $21$ | $4$ | $1^{8}\cdot2^{6}$ |
120.144.9.ivc.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.ivx.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.kjy.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.klh.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.tan.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.tbb.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.tgs.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.thn.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
180.216.13.hq.1 | $180$ | $3$ | $3$ | $13$ | $?$ | not computed |
300.360.21.w.2 | $300$ | $5$ | $5$ | $21$ | $?$ | not computed |