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: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20H3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.72.3.720 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}19&45\\26&13\end{bmatrix}$, $\begin{bmatrix}27&50\\32&33\end{bmatrix}$, $\begin{bmatrix}37&55\\40&59\end{bmatrix}$, $\begin{bmatrix}41&0\\4&47\end{bmatrix}$ |
| 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: | $2^{10}\cdot3^{6}\cdot5^{4}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2$ |
| Newforms: | 360.2.f.c, 3600.2.a.be |
Models
Embedded model Embedded model in $\mathbb{P}^{5}$
| $ 0 $ | $=$ | $ x u + z t $ |
| $=$ | $ - y u + z w$ | |
| $=$ | $x w + y t$ | |
| $=$ | $w^{2} - 3 t^{2} + 2 t u + u^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 3600 x^{4} y^{4} - 1560 x^{4} y^{2} z^{2} + 169 x^{4} z^{4} + 4725 x^{2} y^{4} z^{2} - 1830 x^{2} y^{2} z^{4} + \cdots + 80 z^{8} $ |
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ 15x^{8} + 120x^{6} + 330x^{4} + 600x^{2} + 375 $ |
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:
| $\displaystyle X$ | $=$ | $\displaystyle w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle u$ |
Birational map from embedded model to Weierstrass model:
| $\displaystyle X$ | $=$ | $\displaystyle -\frac{1}{3}wu$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -\frac{160}{3}zt^{4}u^{3}-\frac{440}{9}zt^{3}u^{4}-\frac{400}{27}zt^{2}u^{5}-\frac{40}{27}ztu^{6}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle tu+\frac{1}{3}u^{2}$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{995321925000xz^{9}-879200881875xz^{7}u^{2}-449024684625xz^{5}u^{4}+15205819050xz^{3}u^{6}+33395279055xzu^{8}-248832000000z^{10}+199065195000z^{8}u^{2}+131604793875z^{6}u^{4}+4127389875z^{4}u^{6}-8576009505z^{2}u^{8}+117459742720t^{10}+103884611840t^{9}u-322582744960t^{8}u^{2}-389913207360t^{7}u^{3}+194228280040t^{6}u^{4}+451072683505t^{5}u^{5}+129795286545t^{4}u^{6}-131936703505t^{3}u^{7}-118609009105t^{2}u^{8}-34606140304tu^{9}-4013882632u^{10}}{27000xz^{5}u^{4}-24525xz^{3}u^{6}+6225xzu^{8}+1800z^{4}u^{6}-1395z^{2}u^{8}-79626240t^{10}+31518720t^{9}u+50872320t^{8}u^{2}+1509120t^{7}u^{3}-4327680t^{6}u^{4}-2960t^{5}u^{5}+68160t^{4}u^{6}-13665t^{3}u^{7}+2635t^{2}u^{8}-7tu^{9}+269u^{10}}$ |
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 |
|---|---|---|---|---|---|---|
| 20.36.0.d.2 | $20$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.36.1.bg.1 | $60$ | $2$ | $2$ | $1$ | $1$ | $2$ |
| 60.36.2.fr.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
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.bm.2 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 60.144.5.cq.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 60.144.5.ng.2 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 60.144.5.ni.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 60.144.5.oo.2 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 60.144.5.ot.2 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 60.144.5.qi.2 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 60.144.5.qk.2 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 60.216.15.km.1 | $60$ | $3$ | $3$ | $15$ | $1$ | $1^{6}\cdot2^{3}$ |
| 60.288.17.ho.1 | $60$ | $4$ | $4$ | $17$ | $4$ | $1^{6}\cdot2^{4}$ |
| 60.360.19.rd.1 | $60$ | $5$ | $5$ | $19$ | $5$ | $1^{6}\cdot2^{5}$ |
| 120.144.5.jc.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.sm.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dvm.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dwa.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.edn.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.efa.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.epx.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.eql.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 300.360.19.dd.2 | $300$ | $5$ | $5$ | $19$ | $?$ | not computed |