Invariants
| Level: | $60$ | $\SL_2$-level: | $10$ | Newform level: | $80$ | ||
| 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 | $2^{6}\cdot10^{6}$ | 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: | 10K1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.72.1.369 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}13&30\\24&17\end{bmatrix}$, $\begin{bmatrix}17&35\\46&21\end{bmatrix}$, $\begin{bmatrix}31&35\\26&3\end{bmatrix}$, $\begin{bmatrix}39&5\\20&13\end{bmatrix}$, $\begin{bmatrix}57&5\\40&27\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 60.144.1-60.e.1.1, 60.144.1-60.e.1.2, 60.144.1-60.e.1.3, 60.144.1-60.e.1.4, 60.144.1-60.e.1.5, 60.144.1-60.e.1.6, 60.144.1-60.e.1.7, 60.144.1-60.e.1.8, 120.144.1-60.e.1.1, 120.144.1-60.e.1.2, 120.144.1-60.e.1.3, 120.144.1-60.e.1.4, 120.144.1-60.e.1.5, 120.144.1-60.e.1.6, 120.144.1-60.e.1.7, 120.144.1-60.e.1.8 |
| Cyclic 60-isogeny field degree: | $8$ |
| Cyclic 60-torsion field degree: | $128$ |
| Full 60-torsion field degree: | $30720$ |
Jacobian
| Conductor: | $2^{4}\cdot5$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 80.2.a.b |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 15 x^{2} + 3 z^{2} - z w $ |
| $=$ | $15 y^{2} - 8 z^{2} + 4 z w - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 5 x^{4} + 6 x^{2} z^{2} - 3 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:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}z$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{(704z^{6}-128z^{5}w-80z^{4}w^{2}+20z^{2}w^{4}-8zw^{5}+w^{6})^{3}}{z^{5}(2z-w)^{10}(2z+w)^{2}(3z-w)}$ |
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 |
|---|---|---|---|---|---|---|
| 20.36.1.a.1 | $20$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 30.36.0.c.1 | $30$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.24.1.f.1 | $60$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 60.36.0.b.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 |
|---|---|---|---|---|---|---|
| 60.144.5.by.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 60.144.5.cf.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 60.144.5.cp.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 60.144.5.cr.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 60.144.5.dg.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 60.144.5.di.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 60.144.5.dj.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 60.144.5.dl.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 60.216.13.w.1 | $60$ | $3$ | $3$ | $13$ | $1$ | $1^{6}\cdot2^{3}$ |
| 60.288.13.ge.2 | $60$ | $4$ | $4$ | $13$ | $0$ | $1^{6}\cdot2^{3}$ |
| 60.360.13.e.1 | $60$ | $5$ | $5$ | $13$ | $1$ | $1^{6}\cdot2^{3}$ |
| 120.144.5.no.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.pl.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.sd.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.sr.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.wq.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.xf.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.xl.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ya.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 300.360.13.c.1 | $300$ | $5$ | $5$ | $13$ | $?$ | not computed |