Invariants
| Level: | $60$ | $\SL_2$-level: | $20$ | Newform level: | $180$ | ||
| 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: | $0$ | ||||||
| $\Q$-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: | 60.72.1.332 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}9&10\\10&13\end{bmatrix}$, $\begin{bmatrix}13&50\\42&47\end{bmatrix}$, $\begin{bmatrix}23&30\\54&49\end{bmatrix}$, $\begin{bmatrix}37&35\\36&7\end{bmatrix}$ |
| 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: | $2^{2}\cdot3^{2}\cdot5$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 180.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 5 x^{2} - 3 z w $ |
| $=$ | $5 y^{2} + z^{2} - 2 z w + 5 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 45 x^{4} - 6 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:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 3y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 3w$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{(z^{6}-10z^{5}w+35z^{4}w^{2}-60z^{3}w^{3}+55z^{2}w^{4}-10zw^{5}+5w^{6})^{3}}{w^{10}z^{2}(z-5w)(z-w)^{5}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
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.d.2 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.36.1.x.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 |
|---|---|---|---|---|---|---|
| 60.144.5.bb.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 60.144.5.dk.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 60.144.5.ij.2 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 60.144.5.ip.2 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 60.144.5.ku.2 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 60.144.5.kw.2 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 60.144.5.ky.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 60.144.5.le.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 60.216.13.cy.1 | $60$ | $3$ | $3$ | $13$ | $0$ | $1^{6}\cdot2^{3}$ |
| 60.288.13.ka.1 | $60$ | $4$ | $4$ | $13$ | $0$ | $1^{6}\cdot2^{3}$ |
| 60.360.13.u.1 | $60$ | $5$ | $5$ | $13$ | $1$ | $1^{6}\cdot2^{3}$ |
| 120.144.5.gx.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.xv.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.cnt.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.cpm.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.del.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dez.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dfp.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dhf.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 300.360.13.n.1 | $300$ | $5$ | $5$ | $13$ | $?$ | not computed |