Invariants
| Level: | $60$ | $\SL_2$-level: | $20$ | Newform level: | $900$ | ||
| 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 | $1^{4}\cdot4^{2}\cdot5^{4}\cdot20^{2}$ | Cusp orbits | $2^{2}\cdot4^{2}$ | ||
| Elliptic points: | $0$ 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: | 20H1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.72.1.299 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}23&10\\42&41\end{bmatrix}$, $\begin{bmatrix}33&20\\14&17\end{bmatrix}$, $\begin{bmatrix}43&15\\24&37\end{bmatrix}$, $\begin{bmatrix}57&5\\34&29\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 60-isogeny field degree: | $8$ |
| Cyclic 60-torsion field degree: | $128$ |
| Full 60-torsion field degree: | $30720$ |
Jacobian
| Conductor: | $2^{2}\cdot3^{2}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 900.2.a.b |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ y z - w^{2} $ |
| $=$ | $15 x^{2} - y^{2} - y z - 5 z^{2} - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 5 x^{4} - 15 x^{2} y^{2} + 2 x^{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:
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{y^{18}+30y^{16}w^{2}+405y^{14}w^{4}+3280y^{12}w^{6}+17940y^{10}w^{8}+70680y^{8}w^{10}+208340y^{6}w^{12}+469200y^{4}w^{14}+814350y^{2}w^{16}+125z^{18}+750z^{16}w^{2}+5625z^{14}w^{4}+22000z^{12}w^{6}+82500z^{10}w^{8}+219000z^{8}w^{10}+489700z^{6}w^{12}+850800z^{4}w^{14}+1108950z^{2}w^{16}+1088100w^{18}}{w^{4}z^{2}(z^{2}+w^{2})^{5}(5z^{2}+w^{2})}$ |
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 |
|---|---|---|---|---|---|---|
| 20.36.0.c.1 | $20$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.36.0.b.1 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.36.1.bh.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 |
|---|---|---|---|---|---|---|
| 60.144.5.bv.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.ne.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 60.144.5.nl.1 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 60.144.5.on.2 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 60.144.5.oy.2 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 60.144.5.qg.2 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 60.144.5.qn.2 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 60.216.13.hn.2 | $60$ | $3$ | $3$ | $13$ | $2$ | $1^{6}\cdot2^{3}$ |
| 60.288.13.od.1 | $60$ | $4$ | $4$ | $13$ | $2$ | $1^{6}\cdot2^{3}$ |
| 60.360.13.ch.1 | $60$ | $5$ | $5$ | $13$ | $2$ | $1^{6}\cdot2^{3}$ |
| 120.144.5.jb.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.wu.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dux.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dwu.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ecv.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.egf.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.epi.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.erf.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 300.360.13.bp.1 | $300$ | $5$ | $5$ | $13$ | $?$ | not computed |