Invariants
Level: | $60$ | $\SL_2$-level: | $10$ | Newform level: | $900$ | ||
Index: | $40$ | $\PSL_2$-index: | $40$ | ||||
Genus: | $1 = 1 + \frac{ 40 }{12} - \frac{ 0 }{4} - \frac{ 4 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (none of which are rational) | Cusp widths | $10^{4}$ | Cusp orbits | $4$ | ||
Elliptic points: | $0$ of order $2$ and $4$ 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: | 10H1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.40.1.34 |
Level structure
$\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}5&42\\37&1\end{bmatrix}$, $\begin{bmatrix}41&52\\2&57\end{bmatrix}$, $\begin{bmatrix}57&29\\2&43\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 60-isogeny field degree: | $144$ |
Cyclic 60-torsion field degree: | $2304$ |
Full 60-torsion field degree: | $55296$ |
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 $ | $=$ | $ 2 x^{2} - 3 y^{2} - z^{2} - w^{2} $ |
$=$ | $3 x^{2} + 3 y^{2} + 2 z^{2} - z w$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 9 x^{4} - 42 x^{2} y^{2} + 15 x^{2} z^{2} + 49 y^{4} - 30 y^{2} z^{2} + 5 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:
$\displaystyle X$ | $=$ | $\displaystyle y$ |
$\displaystyle Y$ | $=$ | $\displaystyle x$ |
$\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 40 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -5^4\,\frac{(z+w)^{3}(4z+3w)(4z^{2}+zw+w^{2})^{3}}{(z^{2}-zw-w^{2})^{5}}$ |
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 |
---|---|---|---|---|---|---|
20.20.0.d.1 | $20$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
30.20.1.a.1 | $30$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
60.20.0.a.1 | $60$ | $2$ | $2$ | $0$ | $0$ | 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 |
---|---|---|---|---|---|---|
60.120.5.by.1 | $60$ | $3$ | $3$ | $5$ | $2$ | $1^{4}$ |
60.120.5.cf.1 | $60$ | $3$ | $3$ | $5$ | $2$ | $1^{4}$ |
60.120.7.cz.1 | $60$ | $3$ | $3$ | $7$ | $6$ | $1^{4}\cdot2$ |
60.120.9.dd.1 | $60$ | $3$ | $3$ | $9$ | $6$ | $1^{6}\cdot2$ |
60.160.9.bp.1 | $60$ | $4$ | $4$ | $9$ | $5$ | $1^{8}$ |
60.160.9.bs.1 | $60$ | $4$ | $4$ | $9$ | $6$ | $1^{8}$ |
300.200.9.o.1 | $300$ | $5$ | $5$ | $9$ | $?$ | not computed |