Invariants
| Level: | $60$ | $\SL_2$-level: | $12$ | Newform level: | $1800$ | ||
| Index: | $36$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $1 = 1 + \frac{ 36 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $6^{2}\cdot12^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $4$ of order $2$ and $0$ 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: | 12L1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.36.1.119 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}19&16\\58&29\end{bmatrix}$, $\begin{bmatrix}21&10\\47&39\end{bmatrix}$, $\begin{bmatrix}31&14\\32&17\end{bmatrix}$, $\begin{bmatrix}37&48\\30&11\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 60-isogeny field degree: | $48$ |
| Cyclic 60-torsion field degree: | $768$ |
| Full 60-torsion field degree: | $61440$ |
Jacobian
| Conductor: | $2^{3}\cdot3^{2}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 1800.2.a.m |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 3 y^{2} - 4 z^{2} - 2 z w - w^{2} $ |
| $=$ | $2 x^{2} - 5 z w$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 25 x^{4} + 5 x^{2} z^{2} - 3 y^{2} z^{2} + 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:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{5}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{5}{2}w$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^6\,\frac{(2z^{3}-w^{3})^{3}}{w^{3}z^{6}}$ |
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 |
|---|---|---|---|---|---|---|
| 12.18.0.a.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.18.0.l.1 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.18.1.h.1 | $60$ | $2$ | $2$ | $1$ | $1$ | 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 |
|---|---|---|---|---|---|---|
| 60.72.3.g.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.72.3.ed.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.72.3.fh.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.72.3.fn.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.72.3.jj.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.72.3.jn.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.72.3.jz.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.72.3.kd.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.180.13.lj.1 | $60$ | $5$ | $5$ | $13$ | $7$ | $1^{12}$ |
| 60.216.13.nw.1 | $60$ | $6$ | $6$ | $13$ | $8$ | $1^{12}$ |
| 60.360.25.cah.1 | $60$ | $10$ | $10$ | $25$ | $13$ | $1^{24}$ |
| 120.72.3.kn.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.bao.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.bhk.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.bja.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.cqs.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.cru.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.cva.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.cwc.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 180.108.5.w.1 | $180$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 180.324.21.p.1 | $180$ | $9$ | $9$ | $21$ | $?$ | not computed |