Invariants
| Level: | $60$ | $\SL_2$-level: | $30$ | Newform level: | $3600$ | ||
| Index: | $36$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $3 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $6\cdot30$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 30B3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.36.3.4 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}4&21\\39&37\end{bmatrix}$, $\begin{bmatrix}31&24\\21&13\end{bmatrix}$, $\begin{bmatrix}36&29\\17&39\end{bmatrix}$, $\begin{bmatrix}38&7\\59&35\end{bmatrix}$, $\begin{bmatrix}57&20\\47&39\end{bmatrix}$ |
| 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: | $61440$ |
Jacobian
| Conductor: | $2^{12}\cdot3^{6}\cdot5^{6}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{3}$ |
| Newforms: | 3600.2.a.be, 3600.2.a.e$^{2}$ |
Models
Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ y^{2} t - z w t $ |
| $=$ | $y^{2} w - z w^{2}$ | |
| $=$ | $y^{2} z - z^{2} w$ | |
| $=$ | $y^{3} - y z w$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 125 x^{6} - 22 x^{3} z^{3} + 15 x y^{2} z^{3} + z^{6} $ |
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -15x^{7} + 330x^{4} - 1875x $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Embedded model |
|---|
| $(0:0:0:0:1)$, $(1:0:0:0:0)$ |
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{15}t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Birational map from embedded model to Weierstrass model:
| $\displaystyle X$ | $=$ | $\displaystyle w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -yw^{2}t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle y$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{125x^{2}t^{4}-134103600xzw^{2}t^{2}+3530xwt^{4}+27915840yw^{5}-400308480z^{2}w^{4}-80434944zw^{3}t^{2}+390625w^{2}t^{4}}{w(3300xzwt^{2}-5xt^{4}-16155yw^{4}+231660z^{2}w^{3}-702zw^{2}t^{2})}$ |
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 |
|---|---|---|---|---|---|---|
| 15.18.0.a.1 | $15$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.6.1.b.1 | $60$ | $6$ | $6$ | $1$ | $0$ | $1^{2}$ |
| 60.12.1.n.1 | $60$ | $3$ | $3$ | $1$ | $1$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.72.5.cc.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 60.72.5.cd.1 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 60.72.5.ci.1 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 60.72.5.cj.1 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 60.72.5.cx.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 60.72.5.cx.2 | $60$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 60.72.5.cy.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 60.72.5.cy.2 | $60$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 60.72.5.da.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 60.72.5.da.2 | $60$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 60.72.5.db.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 60.72.5.db.2 | $60$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 60.72.5.dd.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 60.72.5.dd.2 | $60$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 60.72.5.de.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 60.72.5.de.2 | $60$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 60.72.5.dg.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 60.72.5.dg.2 | $60$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 60.72.5.dh.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 60.72.5.dh.2 | $60$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 60.72.5.dm.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 60.72.5.dn.1 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 60.72.5.ds.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 60.72.5.dt.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 60.108.7.j.1 | $60$ | $3$ | $3$ | $7$ | $1$ | $1^{4}$ |
| 60.144.11.z.1 | $60$ | $4$ | $4$ | $11$ | $4$ | $1^{8}$ |
| 60.180.13.fp.1 | $60$ | $5$ | $5$ | $13$ | $4$ | $1^{8}\cdot2$ |
| 120.72.5.zq.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.zt.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bao.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bar.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bda.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bda.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bdd.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bdd.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bdm.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bdm.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bdp.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bdp.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bdy.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bdy.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.beb.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.beb.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bek.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bek.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.ben.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.ben.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bfi.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bfl.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bgg.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bgj.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 180.108.9.d.1 | $180$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 300.180.13.d.1 | $300$ | $5$ | $5$ | $13$ | $?$ | not computed |