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: | 60.72.3.786 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}5&48\\32&19\end{bmatrix}$, $\begin{bmatrix}11&18\\57&25\end{bmatrix}$, $\begin{bmatrix}37&48\\35&49\end{bmatrix}$, $\begin{bmatrix}47&44\\57&35\end{bmatrix}$ |
| 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 $ | $=$ | $ 2 x u + y t - y u $ |
| $=$ | $x z + 2 y w$ | |
| $=$ | $z t - z u - 4 w u$ | |
| $=$ | $5 z w - 5 w^{2} + t u$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 36 x^{4} y^{2} - 3 x^{2} y^{4} + 30 x^{2} y^{2} z^{2} - 75 x^{2} z^{4} + 5 y^{4} z^{2} - 50 y^{2} z^{4} + 125 z^{6} $ |
Geometric Weierstrass model Geometric Weierstrass model
| $ 675 w^{2} $ | $=$ | $ 725 x^{4} - 200 x^{3} y - 415 x^{2} z^{2} + 60 x y z^{2} + 303 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:
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{2}{5}u$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{3125z^{10}+6250z^{8}u^{2}+1875z^{6}u^{4}+250z^{4}u^{6}-425z^{2}u^{8}-3125w^{10}+25000w^{8}u^{2}-67375w^{6}u^{4}+89200w^{4}u^{6}+921390w^{2}u^{8}+16t^{10}-160t^{9}u+720t^{8}u^{2}-1920t^{7}u^{3}+3360t^{6}u^{4}-3263t^{5}u^{5}+3340t^{4}u^{6}+5919t^{3}u^{7}+30040t^{2}u^{8}-36590tu^{9}+16u^{10}}{u^{6}(25w^{4}-50w^{2}u^{2}-t^{2}u^{2}+2tu^{3})}$ |
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.u.1 | $60$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
| 60.36.1.w.1 | $60$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 60.36.1.bf.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.kp.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 60.144.5.kp.2 | $60$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 60.144.5.kq.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 60.144.5.kq.2 | $60$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 60.144.5.nm.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 60.144.5.nm.2 | $60$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 60.144.5.nn.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 60.144.5.nn.2 | $60$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 60.216.15.gg.1 | $60$ | $3$ | $3$ | $15$ | $3$ | $1^{12}$ |
| 60.288.17.ge.1 | $60$ | $4$ | $4$ | $17$ | $6$ | $1^{14}$ |
| 60.360.19.li.1 | $60$ | $5$ | $5$ | $19$ | $6$ | $1^{16}$ |
| 120.144.5.dde.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dde.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ddl.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ddl.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dxa.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dxa.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dxh.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dxh.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 300.360.19.cb.1 | $300$ | $5$ | $5$ | $19$ | $?$ | not computed |