Invariants
Level: | $60$ | $\SL_2$-level: | $30$ | Newform level: | $400$ | ||
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: | 60.72.1.276 |
Level structure
$\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}7&45\\48&17\end{bmatrix}$, $\begin{bmatrix}26&55\\43&43\end{bmatrix}$, $\begin{bmatrix}34&35\\37&46\end{bmatrix}$, $\begin{bmatrix}53&45\\18&11\end{bmatrix}$ |
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: | $2^{4}\cdot5^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 400.2.a.c |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ y^{2} + z^{2} + z w + w^{2} $ |
$=$ | $3 x^{2} - 4 y^{2} + 5 y z - 5 y w + z^{2} + z w + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 25 x^{4} - 15 x^{2} y z - 30 x^{2} z^{2} + 3 y^{2} z^{2} + 9 y z^{3} + 18 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:
$\displaystyle X$ | $=$ | $\displaystyle x$ |
$\displaystyle Y$ | $=$ | $\displaystyle 5w$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{5}{3}y$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 3^3\,\frac{236yz^{17}+170yz^{16}w-13246yz^{15}w^{2}-64540yz^{14}w^{3}-84820yz^{13}w^{4}+161558yz^{12}w^{5}+720362yz^{11}w^{6}+1037528yz^{10}w^{7}+512120yz^{9}w^{8}-512120yz^{8}w^{9}-1037528yz^{7}w^{10}-720362yz^{6}w^{11}-161558yz^{5}w^{12}+84820yz^{4}w^{13}+64540yz^{3}w^{14}+13246yz^{2}w^{15}-170yzw^{16}-236yw^{17}+115z^{18}+2961z^{17}w+12051z^{16}w^{2}-9465z^{15}w^{3}-176940z^{14}w^{4}-494307z^{13}w^{5}-545403z^{12}w^{6}+194823z^{11}w^{7}+1456497z^{10}w^{8}+2105320z^{9}w^{9}+1456497z^{8}w^{10}+194823z^{7}w^{11}-545403z^{6}w^{12}-494307z^{5}w^{13}-176940z^{4}w^{14}-9465z^{3}w^{15}+12051z^{2}w^{16}+2961zw^{17}+115w^{18}}{(z^{2}+zw+w^{2})^{7}(2yz^{3}+3yz^{2}w-3yzw^{2}-2yw^{3}-2z^{4}+5z^{3}w+3z^{2}w^{2}+5zw^{3}-2w^{4})}$ |
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 |
---|---|---|---|---|---|---|
15.36.0.b.2 | $15$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
60.36.0.cg.2 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
60.36.1.fy.1 | $60$ | $2$ | $2$ | $1$ | $0$ | 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 |
---|---|---|---|---|---|---|
60.144.9.cf.2 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
60.144.9.cg.2 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
60.144.9.dg.1 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
60.144.9.dh.1 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
60.144.9.go.2 | $60$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
60.144.9.gp.2 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
60.144.9.hj.2 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
60.144.9.hk.1 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
60.216.9.f.1 | $60$ | $3$ | $3$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
60.288.13.ru.1 | $60$ | $4$ | $4$ | $13$ | $1$ | $1^{6}\cdot2^{3}$ |
60.360.21.cs.1 | $60$ | $5$ | $5$ | $21$ | $6$ | $1^{8}\cdot2^{6}$ |
120.144.9.jgd.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.jgk.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.kxf.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.kxm.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.sar.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.say.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.shd.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.shk.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
180.216.13.ic.2 | $180$ | $3$ | $3$ | $13$ | $?$ | not computed |
300.360.21.g.2 | $300$ | $5$ | $5$ | $21$ | $?$ | not computed |