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: | 60.72.1.407 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}1&55\\43&56\end{bmatrix}$, $\begin{bmatrix}16&45\\27&58\end{bmatrix}$, $\begin{bmatrix}26&55\\17&14\end{bmatrix}$, $\begin{bmatrix}58&25\\29&16\end{bmatrix}$ |
| 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 $ | $=$ | $ 3 x^{2} - z w $ |
| $=$ | $15 y^{2} - 5 z^{2} - 2 z w - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 5 x^{4} + 6 x^{2} z^{2} - 15 y^{2} z^{2} + 9 z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}w$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{(5z^{6}-10z^{3}w^{3}+w^{6})^{3}}{w^{3}z^{15}}$ |
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 |
|---|---|---|---|---|---|---|
| 30.36.0.e.1 | $30$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.36.0.j.1 | $60$ | $2$ | $2$ | $0$ | $0$ | 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.cd.1 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 60.144.9.ci.1 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 60.144.9.df.2 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 60.144.9.dj.1 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 60.144.9.gn.1 | $60$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
| 60.144.9.gr.1 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 60.144.9.hi.2 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 60.144.9.hm.1 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 60.216.9.h.1 | $60$ | $3$ | $3$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 60.288.13.rw.2 | $60$ | $4$ | $4$ | $13$ | $1$ | $1^{6}\cdot2^{3}$ |
| 60.360.21.ct.1 | $60$ | $5$ | $5$ | $21$ | $4$ | $1^{8}\cdot2^{6}$ |
| 120.144.9.jfp.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.jgy.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.kwy.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.kya.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.sak.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.sbm.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.sgw.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.shy.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 180.216.13.id.2 | $180$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 300.360.21.i.1 | $300$ | $5$ | $5$ | $21$ | $?$ | not computed |