Invariants
| Level: | $60$ | $\SL_2$-level: | $12$ | Newform level: | $3600$ | ||
| Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 8 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $6^{4}\cdot12^{4}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $8$ 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: | 12T1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.72.1.86 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}7&26\\32&5\end{bmatrix}$, $\begin{bmatrix}19&45\\12&49\end{bmatrix}$, $\begin{bmatrix}41&12\\46&55\end{bmatrix}$, $\begin{bmatrix}43&6\\32&41\end{bmatrix}$ |
| 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: | $2^{4}\cdot3^{2}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 3600.2.a.v |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 5 x y - 5 y^{2} + w^{2} $ |
| $=$ | $9 x^{2} - 7 x y - 5 y^{2} + z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 25 x^{4} - 3 x^{2} y^{2} + 10 x^{2} z^{2} - 3 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:
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{5}{3}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^6\cdot5^3}\cdot\frac{(125z^{6}+432w^{6})^{3}}{w^{12}z^{6}}$ |
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 |
|---|---|---|---|---|---|---|
| 12.36.0.d.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.36.0.f.1 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.36.1.em.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 |
|---|---|---|---|---|---|---|
| 60.144.5.g.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 60.144.5.cc.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 60.144.5.gn.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 60.144.5.gq.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 60.144.5.ja.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 60.144.5.jd.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 60.144.5.jp.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 60.144.5.jq.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 60.360.25.cfc.1 | $60$ | $5$ | $5$ | $25$ | $11$ | $1^{24}$ |
| 60.432.25.bjw.1 | $60$ | $6$ | $6$ | $25$ | $4$ | $1^{24}$ |
| 60.720.49.ejm.1 | $60$ | $10$ | $10$ | $49$ | $18$ | $1^{48}$ |
| 120.144.5.mi.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ou.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.bxl.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.byg.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.csq.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.cty.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.cwr.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.cwy.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 180.216.9.o.1 | $180$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 180.216.9.w.1 | $180$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 180.216.9.ch.1 | $180$ | $3$ | $3$ | $9$ | $?$ | not computed |