Invariants
| Level: | $60$ | $\SL_2$-level: | $12$ | Newform level: | $144$ | ||
| 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: | $0$ | ||||||
| $\Q$-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: | 60.36.1.161 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}1&2\\44&41\end{bmatrix}$, $\begin{bmatrix}19&32\\2&5\end{bmatrix}$, $\begin{bmatrix}21&26\\20&57\end{bmatrix}$, $\begin{bmatrix}49&52\\41&55\end{bmatrix}$ |
| 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: | $2^{4}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 144.2.a.b |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 10 x^{2} - z w $ |
| $=$ | $15 y^{2} - 4 z^{2} + 2 z w - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 5 x^{2} z^{2} - 15 y^{2} z^{2} + 25 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:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{10}w$ |
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 |
|---|---|---|---|---|---|---|
| 12.18.1.g.1 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.18.0.a.1 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.18.0.k.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 |
|---|---|---|---|---|---|---|
| 60.72.3.br.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.72.3.cp.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.72.3.hf.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.72.3.hg.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.72.3.jo.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.72.3.js.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.72.3.ke.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.72.3.ki.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.180.13.lk.1 | $60$ | $5$ | $5$ | $13$ | $3$ | $1^{12}$ |
| 60.216.13.nx.1 | $60$ | $6$ | $6$ | $13$ | $5$ | $1^{12}$ |
| 60.360.25.cai.1 | $60$ | $10$ | $10$ | $25$ | $6$ | $1^{24}$ |
| 120.72.3.ko.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.qs.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.buw.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.bvd.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.csb.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.ctd.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.cwj.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.cxl.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 180.108.5.r.1 | $180$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 180.324.21.q.1 | $180$ | $9$ | $9$ | $21$ | $?$ | not computed |