Invariants
| Level: | $60$ | $\SL_2$-level: | $20$ | Newform level: | $360$ | ||
| Index: | $144$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $5 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{4}\cdot10^{4}\cdot20^{4}$ | Cusp orbits | $2^{4}\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20I5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.144.5.543 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}9&40\\8&11\end{bmatrix}$, $\begin{bmatrix}31&30\\36&19\end{bmatrix}$, $\begin{bmatrix}31&40\\40&53\end{bmatrix}$, $\begin{bmatrix}37&15\\38&29\end{bmatrix}$ |
| 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: | $15360$ |
Jacobian
| Conductor: | $2^{13}\cdot3^{8}\cdot5^{5}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{3}\cdot2$ |
| Newforms: | 40.2.a.a, 180.2.a.a$^{2}$, 360.2.f.c |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ x^{2} - x z + y^{2} + z^{2} $ |
| $=$ | $3 x^{2} + x z - x w - y^{2} - z^{2} + w^{2} + t^{2}$ | |
| $=$ | $x^{2} - x z + x t - 4 y^{2} + z^{2} - 2 w t + t^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 25 x^{8} - 50 x^{6} z^{2} - 15 x^{4} y^{2} z^{2} + 35 x^{4} z^{4} + 24 x^{2} y^{2} z^{4} - 10 x^{2} z^{6} + \cdots + z^{8} $ |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=11$, and therefore no rational points.
Maps between models of this curve
Birational map from canonical model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle t$ |
Maps to other modular curves
$j$-invariant map of degree 144 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{3^3}{2^{20}}\cdot\frac{190153669921875xw^{17}+1092128568750000xw^{16}t+2020414387500000xw^{15}t^{2}+2123817975000000xw^{14}t^{3}+2682830868750000xw^{13}t^{4}-83808972000000xw^{12}t^{5}-4223362356000000xw^{11}t^{6}-1960180819200000xw^{10}t^{7}-1195451676000000xw^{9}t^{8}-315685516800000xw^{8}t^{9}+1296232657920000xw^{7}t^{10}+2112434176000xw^{6}t^{11}+294218376192000xw^{5}t^{12}-20784562176000xw^{4}t^{13}-112117628928000xw^{3}t^{14}+39532851363840xw^{2}t^{15}-26841029345280xwt^{16}+7060061159424xt^{17}-39017162109375w^{18}-370619043750000w^{17}t-789015779296875w^{16}t^{2}+106880006250000w^{15}t^{3}+1306773126562500w^{14}t^{4}+1647663714000000w^{13}t^{5}+2895454329750000w^{12}t^{6}+1147019486400000w^{11}t^{7}-1030318477200000w^{10}t^{8}-352787750400000w^{9}t^{9}-1047746288160000w^{8}t^{10}-206607506432000w^{7}t^{11}+182885541120000w^{6}t^{12}-42928005120000w^{5}t^{13}+152469660672000w^{4}t^{14}-36153640878080w^{3}t^{15}+17489099489280w^{2}t^{16}-10928057745408wt^{17}+706390851584t^{18}}{t^{10}(6024375xw^{7}+6777000xw^{6}t-17091000xw^{5}t^{2}+252000xw^{4}t^{3}+3996000xw^{3}t^{4}-418560xw^{2}t^{5}-84480xwt^{6}+3584xt^{7}+658125w^{8}-189000w^{7}t+5855625w^{6}t^{2}+2547000w^{5}t^{3}-5926500w^{4}t^{4}+606720w^{3}t^{5}+408480w^{2}t^{6}-32128wt^{7}-1856t^{8})}$ |
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 |
|---|---|---|---|---|---|---|
| 20.72.1.t.1 | $20$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 30.72.1.g.2 | $30$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 60.72.1.bv.1 | $60$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 60.72.3.of.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.72.3.om.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.72.3.ov.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 60.72.3.za.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.288.9.by.2 | $60$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 60.288.9.by.3 | $60$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 60.288.9.bz.1 | $60$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 60.288.9.bz.4 | $60$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 60.432.29.cjx.2 | $60$ | $3$ | $3$ | $29$ | $0$ | $1^{12}\cdot2^{6}$ |
| 60.576.33.lk.1 | $60$ | $4$ | $4$ | $33$ | $0$ | $1^{14}\cdot2^{7}$ |
| 60.720.37.mv.1 | $60$ | $5$ | $5$ | $37$ | $5$ | $1^{16}\cdot2^{8}$ |
| 120.288.9.kq.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9.kq.3 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9.kr.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9.kr.4 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.17.bapp.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.288.17.bapt.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.288.17.bsel.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.288.17.bsen.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.288.17.cfim.2 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.288.17.cfin.2 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.288.17.cfiq.2 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.288.17.cfir.2 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.288.17.cfiu.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.288.17.cfiv.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.288.17.cfiy.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.288.17.cfiz.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.288.17.cmhp.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.288.17.cmhr.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.288.17.cmmn.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.288.17.cmmr.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |