Invariants
| Level: | $60$ | $\SL_2$-level: | $20$ | Newform level: | $3600$ | ||
| Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $3 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot10^{2}\cdot20^{2}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20J3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.72.3.787 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}3&56\\20&33\end{bmatrix}$, $\begin{bmatrix}19&38\\50&33\end{bmatrix}$, $\begin{bmatrix}33&52\\55&49\end{bmatrix}$, $\begin{bmatrix}49&42\\0&13\end{bmatrix}$ |
| 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: | $2^{11}\cdot3^{4}\cdot5^{5}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{3}$ |
| Newforms: | 200.2.a.c, 720.2.a.h, 3600.2.a.be |
Models
Embedded model Embedded model in $\mathbb{P}^{5}$
| $ 0 $ | $=$ | $ - x u + 2 w t $ |
| $=$ | $x y + x z - y w - 3 z w$ | |
| $=$ | $2 y t - y u + 2 z t - 3 z u$ | |
| $=$ | $3 x^{2} - 6 x w + y^{2} - 2 y z + 5 z^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 108 x^{6} y^{2} - 2025 x^{4} y^{4} - 270 x^{4} y^{2} z^{2} - 9 x^{4} z^{4} + 4500 x^{2} y^{4} z^{2} + \cdots - 25 z^{8} $ |
Geometric Weierstrass model Geometric Weierstrass model
| $ 25 w^{2} $ | $=$ | $ -25 x^{4} + 15 x^{2} z^{2} + 9 z^{4} $ |
| $0$ | $=$ | $-2 x^{2} + 2 x y - 3 y^{2} + z^{2}$ |
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:
| $\displaystyle X$ | $=$ | $\displaystyle w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{2}{5}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{5}u$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2}{3^2}\cdot\frac{874800000xw^{9}-116640000xw^{7}u^{2}-358547472000xw^{5}u^{4}-7994469600xw^{3}u^{6}+1175245920xwu^{8}-874800000w^{10}+145800000w^{8}u^{2}+597692520000w^{6}u^{4}+19957716000w^{4}u^{6}-1840218000w^{2}u^{8}-18662400t^{10}+454118400t^{9}u-4234809600t^{8}u^{2}+19042560000t^{7}u^{3}-43921584000t^{6}u^{4}+55328572800t^{5}u^{5}-37398928800t^{4}u^{6}+11116859200t^{3}u^{7}+81549500t^{2}u^{8}-318586196tu^{9}-142938421u^{10}}{54000xw^{5}u^{4}-7200xw^{3}u^{6}-165xwu^{8}-54000w^{6}u^{4}+9000w^{4}u^{6}-45w^{2}u^{8}-583200t^{10}+3499200t^{9}u-9136800t^{8}u^{2}+13635000t^{7}u^{3}-12831750t^{6}u^{4}+7910550t^{5}u^{5}-3200300t^{4}u^{6}+820700t^{3}u^{7}-121250t^{2}u^{8}+7904tu^{9}-21u^{10}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 20.36.1.g.1 | $20$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 60.12.0.s.1 | $60$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
| 60.36.1.v.1 | $60$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 60.36.1.bg.1 | $60$ | $2$ | $2$ | $1$ | $1$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.144.5.kl.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 60.144.5.kl.2 | $60$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 60.144.5.km.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 60.144.5.km.2 | $60$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 60.144.5.ni.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 60.144.5.ni.2 | $60$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 60.144.5.nj.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 60.144.5.nj.2 | $60$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 60.216.15.ge.1 | $60$ | $3$ | $3$ | $15$ | $3$ | $1^{12}$ |
| 60.288.17.gc.1 | $60$ | $4$ | $4$ | $17$ | $6$ | $1^{14}$ |
| 60.360.19.kw.1 | $60$ | $5$ | $5$ | $19$ | $6$ | $1^{16}$ |
| 120.144.5.dcc.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dcc.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dcj.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dcj.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dvy.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dvy.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dwf.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dwf.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 300.360.19.bz.1 | $300$ | $5$ | $5$ | $19$ | $?$ | not computed |