Invariants
| Level: | $60$ | $\SL_2$-level: | $30$ | Newform level: | $20$ | ||
| 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: | 60.72.1.411 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}1&50\\19&19\end{bmatrix}$, $\begin{bmatrix}7&50\\17&17\end{bmatrix}$, $\begin{bmatrix}17&35\\46&17\end{bmatrix}$, $\begin{bmatrix}19&40\\25&17\end{bmatrix}$ |
| 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: | $2^{2}\cdot5$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 20.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 15 x^{2} + z w $ |
| $=$ | $15 y^{2} - 5 z^{2} - 2 z w - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 6 x^{2} z^{2} - 3 y^{2} z^{2} + 45 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:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{15}w$ |
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 |
|---|---|---|---|---|---|---|
| 30.36.1.q.1 | $30$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.36.0.j.1 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.36.0.cg.2 | $60$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 60.144.9.bc.2 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 60.144.9.bf.2 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 60.144.9.cn.1 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 60.144.9.cr.1 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 60.144.9.ix.1 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 60.144.9.iz.1 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 60.144.9.je.2 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 60.144.9.jg.2 | $60$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
| 60.216.9.bi.2 | $60$ | $3$ | $3$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 60.288.13.sl.2 | $60$ | $4$ | $4$ | $13$ | $1$ | $1^{6}\cdot2^{3}$ |
| 60.360.21.db.1 | $60$ | $5$ | $5$ | $21$ | $4$ | $1^{8}\cdot2^{6}$ |
| 120.144.9.ixg.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.iyb.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.jth.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.juj.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.tcr.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.tdf.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.tev.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.tfj.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 180.216.13.hn.1 | $180$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 300.360.21.t.2 | $300$ | $5$ | $5$ | $21$ | $?$ | not computed |