Invariants
| Level: | $60$ | $\SL_2$-level: | $20$ | Newform level: | $400$ | ||
| 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: | $0$ | ||||||
| $\Q$-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: | 60.72.1.390 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}13&5\\54&13\end{bmatrix}$, $\begin{bmatrix}13&25\\44&31\end{bmatrix}$, $\begin{bmatrix}19&45\\36&31\end{bmatrix}$, $\begin{bmatrix}41&35\\14&9\end{bmatrix}$ |
| 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: | $2^{4}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 400.2.a.c |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} - 2 x w - 3 y^{2} + 5 w^{2} $ |
| $=$ | $5 x w + 3 z^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 5 x^{4} + 6 x^{2} z^{2} - 15 y^{2} z^{2} + 9 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:
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{5}{3}w$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 3^3\,\frac{2916xy^{16}w-25920xy^{14}w^{3}+95472xy^{12}w^{5}-186624xy^{10}w^{7}+203520xy^{8}w^{9}-113152xy^{6}w^{11}+15360xy^{4}w^{13}+12288xy^{2}w^{15}-4096xw^{17}-729y^{18}-5832y^{16}w^{2}+103680y^{14}w^{4}-521856y^{12}w^{6}+1365120y^{10}w^{8}-2111232y^{8}w^{10}+1986304y^{6}w^{12}-1105920y^{4}w^{14}+331776y^{2}w^{16}-40960w^{18}}{w^{10}(3y^{2}-4w^{2})^{2}(6xy^{2}w-4xw^{3}-9y^{4}+39y^{2}w^{2}-40w^{4})}$ |
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 |
|---|---|---|---|---|---|---|
| 20.36.1.e.1 | $20$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.36.0.b.2 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.36.0.c.2 | $60$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 60.144.5.bq.2 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 60.144.5.cp.2 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 60.144.5.hs.2 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 60.144.5.hx.2 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 60.144.5.pj.1 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 60.144.5.pm.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 60.144.5.pt.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 60.144.5.pv.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 60.216.13.gb.1 | $60$ | $3$ | $3$ | $13$ | $0$ | $1^{6}\cdot2^{3}$ |
| 60.288.13.nn.2 | $60$ | $4$ | $4$ | $13$ | $1$ | $1^{6}\cdot2^{3}$ |
| 60.360.13.bz.1 | $60$ | $5$ | $5$ | $13$ | $1$ | $1^{6}\cdot2^{3}$ |
| 120.144.5.ij.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.se.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.cjj.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.cks.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ejb.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ejw.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.elv.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.emf.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 300.360.13.z.2 | $300$ | $5$ | $5$ | $13$ | $?$ | not computed |