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: | 60.72.1.308 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}7&15\\18&53\end{bmatrix}$, $\begin{bmatrix}11&0\\46&1\end{bmatrix}$, $\begin{bmatrix}13&20\\28&27\end{bmatrix}$, $\begin{bmatrix}31&10\\20&7\end{bmatrix}$ |
| 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 $ | $=$ | $ 5 x z + w^{2} $ |
| $=$ | $x^{2} - 2 x z - 3 y^{2} + 5 z^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 15 x^{2} y^{2} + 2 x^{2} z^{2} + 5 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:
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{5}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{5}w$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{3^2}{5}\cdot\frac{4271484375y^{18}+3417187500y^{16}w^{2}+455625000y^{14}w^{4}-344250000y^{12}w^{6}-143775000y^{10}w^{8}-19980000y^{8}w^{10}-894000y^{6}w^{12}-187200y^{4}w^{14}-135960y^{2}w^{16}-423828125000z^{18}-508593750000z^{16}w^{2}-274609375000z^{14}w^{4}-88937500000z^{12}w^{6}-19437500000z^{10}w^{8}-3052500000z^{8}w^{10}-354900000z^{6}w^{12}-30220000z^{4}w^{14}-1646000z^{2}w^{16}-57568w^{18}}{w^{4}z^{2}(5z^{2}+w^{2})^{5}(25z^{2}+w^{2})}$ |
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.2 | $20$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.36.0.b.2 | $60$ | $2$ | $2$ | $0$ | $0$ | 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.2 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 60.144.5.dg.2 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 60.144.5.ne.2 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 60.144.5.nl.2 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 60.144.5.on.1 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 60.144.5.oy.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 60.144.5.qg.1 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 60.144.5.qn.1 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 60.216.13.hn.1 | $60$ | $3$ | $3$ | $13$ | $2$ | $1^{6}\cdot2^{3}$ |
| 60.288.13.od.2 | $60$ | $4$ | $4$ | $13$ | $2$ | $1^{6}\cdot2^{3}$ |
| 60.360.13.ch.1 | $60$ | $5$ | $5$ | $13$ | $2$ | $1^{6}\cdot2^{3}$ |
| 120.144.5.jb.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.wu.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dux.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dwu.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ecv.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.egf.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.epi.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.erf.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 300.360.13.bp.2 | $300$ | $5$ | $5$ | $13$ | $?$ | not computed |