| $\GL_2(\Z/60\Z)$-generators: |
$\begin{bmatrix}17&16\\33&7\end{bmatrix}$, $\begin{bmatrix}23&10\\12&17\end{bmatrix}$, $\begin{bmatrix}41&6\\21&59\end{bmatrix}$, $\begin{bmatrix}49&50\\15&31\end{bmatrix}$, $\begin{bmatrix}53&52\\15&31\end{bmatrix}$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
60.96.2-60.f.1.1, 60.96.2-60.f.1.2, 60.96.2-60.f.1.3, 60.96.2-60.f.1.4, 60.96.2-60.f.1.5, 60.96.2-60.f.1.6, 60.96.2-60.f.1.7, 60.96.2-60.f.1.8, 60.96.2-60.f.1.9, 60.96.2-60.f.1.10, 60.96.2-60.f.1.11, 60.96.2-60.f.1.12, 60.96.2-60.f.1.13, 60.96.2-60.f.1.14, 60.96.2-60.f.1.15, 60.96.2-60.f.1.16, 120.96.2-60.f.1.1, 120.96.2-60.f.1.2, 120.96.2-60.f.1.3, 120.96.2-60.f.1.4, 120.96.2-60.f.1.5, 120.96.2-60.f.1.6, 120.96.2-60.f.1.7, 120.96.2-60.f.1.8, 120.96.2-60.f.1.9, 120.96.2-60.f.1.10, 120.96.2-60.f.1.11, 120.96.2-60.f.1.12, 120.96.2-60.f.1.13, 120.96.2-60.f.1.14, 120.96.2-60.f.1.15, 120.96.2-60.f.1.16, 120.96.2-60.f.1.17, 120.96.2-60.f.1.18, 120.96.2-60.f.1.19, 120.96.2-60.f.1.20, 120.96.2-60.f.1.21, 120.96.2-60.f.1.22, 120.96.2-60.f.1.23, 120.96.2-60.f.1.24, 120.96.2-60.f.1.25, 120.96.2-60.f.1.26, 120.96.2-60.f.1.27, 120.96.2-60.f.1.28, 120.96.2-60.f.1.29, 120.96.2-60.f.1.30, 120.96.2-60.f.1.31, 120.96.2-60.f.1.32, 120.96.2-60.f.1.33, 120.96.2-60.f.1.34, 120.96.2-60.f.1.35, 120.96.2-60.f.1.36, 120.96.2-60.f.1.37, 120.96.2-60.f.1.38, 120.96.2-60.f.1.39, 120.96.2-60.f.1.40, 120.96.2-60.f.1.41, 120.96.2-60.f.1.42, 120.96.2-60.f.1.43, 120.96.2-60.f.1.44, 120.96.2-60.f.1.45, 120.96.2-60.f.1.46, 120.96.2-60.f.1.47, 120.96.2-60.f.1.48 |
| Cyclic 60-isogeny field degree: |
$12$ |
| Cyclic 60-torsion field degree: |
$192$ |
| Full 60-torsion field degree: |
$46080$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 8 x^{3} + x^{2} y - 2 x^{2} z + x^{2} w + 8 x y^{2} - 2 x y z + 2 x y w + 2 x z^{2} - 2 x z w + \cdots - y w^{2} $ |
| $=$ | $4 x^{3} + 7 x^{2} y + 7 x^{2} z + x^{2} w + 2 x y^{2} - 6 x y z + 4 x y w - x z^{2} - 2 x z w + \cdots + 2 z^{3}$ |
| $=$ | $4 x^{3} + 7 x^{2} y - 9 x^{2} z + x^{2} w + 2 x y^{2} + 7 x y z + 4 x y w + 3 x z^{2} - x z w + \cdots - 2 z^{2} w$ |
| $=$ | $3 x^{2} y - x^{2} w - 9 x y^{2} - 7 x y z - 7 x y w + x z w + 3 y^{3} - 7 y^{2} z + 4 y^{2} w + \cdots + 2 z^{2} w$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 90 x^{5} - 90 x^{4} y + 960 x^{4} z + 3 x^{3} y^{2} - 90 x^{3} y z + 2260 x^{3} z^{2} + \cdots - 45 y^{3} z^{2} $ |
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Weierstrass model Weierstrass model
| $ y^{2} + y $ | $=$ | $ 36x^{6} - 108x^{5} + 195x^{4} - 210x^{3} + 130x^{2} - 43x - 10 $ |
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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.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 2z$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{5}w$ |
Birational map from embedded model to Weierstrass model:
| $\displaystyle X$ |
$=$ |
$\displaystyle \frac{11}{155}y^{5}-\frac{4}{465}y^{4}z-\frac{1}{72075}y^{4}w-\frac{112}{216225}y^{3}zw+\frac{7306}{360375}y^{3}w^{2}-\frac{128}{72075}y^{2}z^{2}w-\frac{8}{4805}y^{2}zw^{2}-\frac{6266}{1081125}y^{2}w^{3}+\frac{48}{120125}yzw^{3}+\frac{2123}{1801875}yw^{4}-\frac{128}{120125}z^{2}w^{3}-\frac{4}{120125}zw^{4}+\frac{1}{120125}w^{5}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{19568}{11171625}y^{15}-\frac{216}{3723875}y^{14}z+\frac{6402646}{5194805625}y^{14}w+\frac{296}{1340595}y^{13}z^{2}-\frac{488308}{5194805625}y^{13}zw+\frac{6695526416}{2415584615625}y^{13}w^{2}+\frac{6568}{41558445}y^{12}z^{2}w-\frac{144886316}{805194871875}y^{12}zw^{2}+\frac{1790814062078}{1123246846265625}y^{12}w^{3}+\frac{191878768}{483116923125}y^{11}z^{2}w^{2}-\frac{11153358584}{74883123084375}y^{11}zw^{3}+\frac{27372003376048}{16848702693984375}y^{11}w^{4}+\frac{773630444144}{3369740538796875}y^{10}z^{2}w^{3}-\frac{2400074823808}{16848702693984375}y^{10}zw^{4}+\frac{176880420862814}{252730540409765625}y^{10}w^{5}+\frac{1386828882472}{5616234231328125}y^{9}z^{2}w^{4}-\frac{5596603087492}{84243513469921875}y^{9}zw^{5}+\frac{482724086028272}{1263652702048828125}y^{9}w^{6}+\frac{983352044264}{9360390385546875}y^{8}z^{2}w^{5}-\frac{1085621499916}{28081171156640625}y^{8}zw^{6}+\frac{124791900369374}{1263652702048828125}y^{8}w^{7}+\frac{566183392}{9740260546875}y^{7}z^{2}w^{6}-\frac{215856536048}{28081171156640625}y^{7}zw^{7}+\frac{14419047872912}{702029278916015625}y^{7}w^{8}+\frac{74152767904}{5200216880859375}y^{6}z^{2}w^{7}-\frac{135547174168}{46801951927734375}y^{6}zw^{8}-\frac{19743365018}{26001084404296875}y^{6}w^{9}+\frac{41036357288}{15600650642578125}y^{5}z^{2}w^{8}+\frac{143311385596}{234009759638671875}y^{5}zw^{9}-\frac{751286650576}{390016266064453125}y^{5}w^{10}-\frac{17590101704}{78003253212890625}y^{4}z^{2}w^{9}+\frac{5937753092}{78003253212890625}y^{4}zw^{10}-\frac{3358050406}{8667028134765625}y^{4}w^{11}-\frac{8191549552}{26001084404296875}y^{3}z^{2}w^{10}+\frac{36716730904}{390016266064453125}y^{3}zw^{11}-\frac{224818164208}{5850243990966796875}y^{3}w^{12}-\frac{1791606256}{43335140673828125}y^{2}z^{2}w^{11}+\frac{13640624}{43335140673828125}y^{2}zw^{12}-\frac{117435922}{234009759638671875}y^{2}w^{13}-\frac{731112}{1733405626953125}yz^{2}w^{12}-\frac{42292}{8667028134765625}yzw^{13}-\frac{59536}{26001084404296875}yw^{14}-\frac{1944}{1733405626953125}z^{2}w^{13}-\frac{36}{1733405626953125}zw^{14}-\frac{6}{1733405626953125}w^{15}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{2}{155}y^{5}-\frac{4}{155}y^{4}z-\frac{1832}{72075}y^{4}w-\frac{112}{72075}y^{3}zw+\frac{716}{1081125}y^{3}w^{2}-\frac{128}{24025}y^{2}z^{2}w-\frac{24}{4805}y^{2}zw^{2}-\frac{392}{24025}y^{2}w^{3}+\frac{144}{120125}yzw^{3}-\frac{1894}{1801875}yw^{4}-\frac{384}{120125}z^{2}w^{3}-\frac{12}{120125}zw^{4}$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{3^3\cdot5^2}{2^{15}}\cdot\frac{519464769046875xyz^{8}-2830165247025000xyz^{7}w+3007521258750000xyz^{6}w^{2}+194160066948000xyz^{5}w^{3}-565500500892000xyz^{4}w^{4}+4021313305641600xyz^{3}w^{5}-3819897871161600xyz^{2}w^{6}-196737022282240xyzw^{7}-15847605759232xyw^{8}-50652970312500xz^{9}-21992133065625xz^{8}w+325117533915000xz^{7}w^{2}+728009094006000xz^{6}w^{3}-1532934903535200xz^{5}w^{4}+207920575706400xz^{4}w^{5}+137764982459520xz^{3}w^{6}-8531744144640xz^{2}w^{7}+1278942521856xzw^{8}-159033917696xw^{9}+301164764765625y^{2}z^{8}-2077254468225000y^{2}z^{7}w+7002136605870000y^{2}z^{6}w^{2}-13859510368716000y^{2}z^{5}w^{3}+14743357869276000y^{2}z^{4}w^{4}-8047916457302400y^{2}z^{3}w^{5}+2153700893203200y^{2}z^{2}w^{6}+79402106293760y^{2}zw^{7}+4568507431168y^{2}w^{8}-50652970312500yz^{9}-98212826615625yz^{8}w+247556724525000yz^{7}w^{2}+1398990228858000yz^{6}w^{3}-2275523888839200yz^{5}w^{4}+968039585565600yz^{4}w^{5}-838216509617280yz^{3}w^{6}+408323047420160yz^{2}w^{7}+23353670981120yzw^{8}+2163281667840yw^{9}-28398878437500z^{10}+63913303631250z^{9}w+808352133570000z^{8}w^{2}-2634418286172000z^{7}w^{3}+3023700911150400z^{6}w^{4}-2461041012801600z^{5}w^{5}+1116267814920960z^{4}w^{6}+101281960727040z^{3}w^{7}+7091645442048z^{2}w^{8}+318067835392zw^{9}}{2025000xyz^{8}-10665000xyz^{7}w-4286250xyz^{6}w^{2}+27261000xyz^{5}w^{3}-6103906875xyz^{4}w^{4}+12367840800xyz^{3}w^{5}+3589528650xyz^{2}w^{6}-14586397480xyzw^{7}+3641399060xyw^{8}+405000xz^{8}w-405000xz^{7}w^{2}-5393250xz^{6}w^{3}-1431000xz^{5}w^{4}-1222941375xz^{4}w^{5}+2474085600xz^{3}w^{6}-592785150xz^{2}w^{7}-291835560xzw^{8}+25957444xw^{9}-540000y^{2}z^{7}w+1653750y^{2}z^{6}w^{2}+2151000y^{2}z^{5}w^{3}-6151145625y^{2}z^{4}w^{4}+24601630800y^{2}z^{3}w^{5}-33423959250y^{2}z^{2}w^{6}+16396892120y^{2}zw^{7}-2007890180y^{2}w^{8}+1485000yz^{8}w-6210000yz^{7}w^{2}-2540250yz^{6}w^{3}+8181000yz^{5}w^{4}-1219017975yz^{4}w^{5}+3295521600yz^{3}w^{6}-3052251190yz^{2}w^{7}+1783748360yzw^{8}-423259836yw^{9}-810000z^{9}w+3645000z^{8}w^{2}+3496500z^{7}w^{3}-7735500z^{6}w^{4}+2442642750z^{5}w^{5}-7407963000z^{4}w^{6}+6108015900z^{3}w^{7}-973775520z^{2}w^{8}-51914888zw^{9}}$ |
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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.
This modular curve is minimally covered by the modular curves in the database listed below.
