$\GL_2(\Z/60\Z)$-generators: |
$\begin{bmatrix}3&35\\4&19\end{bmatrix}$, $\begin{bmatrix}31&5\\0&19\end{bmatrix}$, $\begin{bmatrix}31&30\\24&1\end{bmatrix}$, $\begin{bmatrix}33&35\\40&9\end{bmatrix}$, $\begin{bmatrix}51&25\\4&53\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
60.144.1-60.m.2.1, 60.144.1-60.m.2.2, 60.144.1-60.m.2.3, 60.144.1-60.m.2.4, 60.144.1-60.m.2.5, 60.144.1-60.m.2.6, 60.144.1-60.m.2.7, 60.144.1-60.m.2.8, 120.144.1-60.m.2.1, 120.144.1-60.m.2.2, 120.144.1-60.m.2.3, 120.144.1-60.m.2.4, 120.144.1-60.m.2.5, 120.144.1-60.m.2.6, 120.144.1-60.m.2.7, 120.144.1-60.m.2.8, 120.144.1-60.m.2.9, 120.144.1-60.m.2.10, 120.144.1-60.m.2.11, 120.144.1-60.m.2.12, 120.144.1-60.m.2.13, 120.144.1-60.m.2.14, 120.144.1-60.m.2.15, 120.144.1-60.m.2.16, 120.144.1-60.m.2.17, 120.144.1-60.m.2.18, 120.144.1-60.m.2.19, 120.144.1-60.m.2.20, 120.144.1-60.m.2.21, 120.144.1-60.m.2.22, 120.144.1-60.m.2.23, 120.144.1-60.m.2.24 |
Cyclic 60-isogeny field degree: |
$4$ |
Cyclic 60-torsion field degree: |
$64$ |
Full 60-torsion field degree: |
$30720$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 5 x^{2} - 3 x y - x z + 2 x w + y^{2} + 2 y z - 4 y w $ |
| $=$ | $x y - 8 x z + x w - y z + 2 y w + 8 z^{2} - 2 z w + 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 256 x^{4} + 54 x^{3} y - 112 x^{3} z + 3 x^{2} y^{2} - 66 x^{2} y z + 42 x^{2} z^{2} - 6 x y^{2} z + \cdots + z^{4} $ |
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 4w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle 2y$ |
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 -3\,\frac{2946279375000xz^{17}-31058527124025xz^{16}w+104613516111150xz^{15}w^{2}-122571439353600xz^{14}w^{3}-68315720802750xz^{13}w^{4}+340290410842800xz^{12}w^{5}-395882429253450xz^{11}w^{6}+233146750442700xz^{10}w^{7}-73577535179175xz^{9}w^{8}+10335347297250xz^{8}w^{9}+343034450025xz^{7}w^{10}-307585583100xz^{6}w^{11}+36873803100xz^{5}w^{12}-476799900xz^{4}w^{13}-299183625xz^{3}w^{14}+3580950xz^{2}w^{15}+2343750xzw^{16}-156125063802y^{2}z^{16}+617407968912y^{2}z^{15}w+1522177586802y^{2}z^{14}w^{2}-12046808799108y^{2}z^{13}w^{3}+26205939961188y^{2}z^{12}w^{4}-27864878457264y^{2}z^{11}w^{5}+15025222890228y^{2}z^{10}w^{6}-2814690420084y^{2}z^{9}w^{7}-1021686717234y^{2}z^{8}w^{8}+653401286484y^{2}z^{7}w^{9}-126753155142y^{2}z^{6}w^{10}+8118572574y^{2}z^{5}w^{11}+121523163y^{2}z^{4}w^{12}-13493592y^{2}z^{3}w^{13}+2675652y^{2}z^{2}w^{14}-247242y^{2}zw^{15}-5187y^{2}w^{16}+156125063802yz^{17}-1080758951001yz^{16}w+4949516377452yz^{15}w^{2}-14038236430218yz^{14}w^{3}+14749926258486yz^{13}w^{4}+17148962187990yz^{12}w^{5}-62229906204846yz^{11}w^{6}+69326202477330yz^{10}w^{7}-38540436280599yz^{9}w^{8}+10518722427093yz^{8}w^{9}-745891296165yz^{7}w^{10}-276464619303yz^{6}w^{11}+64162782630yz^{5}w^{12}-4253847552yz^{4}w^{13}+42186999yz^{3}w^{14}+5908821yz^{2}w^{15}-408612yzw^{16}-10374yw^{17}-3570779630213z^{18}+33835384917960z^{17}w-105234011453226z^{16}w^{2}+109136253291178z^{15}w^{3}+101646179814732z^{14}w^{4}-407536522439772z^{13}w^{5}+524627055574404z^{12}w^{6}-400918549729848z^{11}w^{7}+205652393309976z^{10}w^{8}-71151411889654z^{9}w^{9}+14671272455304z^{8}w^{10}-904056064908z^{7}w^{11}-312224923524z^{6}w^{12}+76147873818z^{5}w^{13}-5835343932z^{4}w^{14}-48205082z^{3}w^{15}+26564436z^{2}w^{16}+776130zw^{17}-83312w^{18}}{z(8437500xz^{16}+208828125xz^{15}w-878203125xz^{14}w^{2}+2644921875xz^{12}w^{4}-1679908275xz^{11}w^{5}-1523254125xz^{10}w^{6}+1155427800xz^{9}w^{7}+297891675xz^{8}w^{8}-218347500xz^{7}w^{9}-30776625xz^{6}w^{10}+12988800xz^{5}w^{11}+1583400xz^{4}w^{12}-168000xz^{3}w^{13}-18000xz^{2}w^{14}+1687500y^{2}z^{15}-14484375y^{2}z^{14}w+102187500y^{2}z^{12}w^{3}-96146164y^{2}z^{11}w^{4}-95631857y^{2}z^{10}w^{5}+110634800y^{2}z^{9}w^{6}+24109700y^{2}z^{8}w^{7}-33878785y^{2}z^{7}w^{8}-3239807y^{2}z^{6}w^{9}+3535984y^{2}z^{5}w^{10}+333730y^{2}z^{4}w^{11}-107280y^{2}z^{3}w^{12}-11040y^{2}z^{2}w^{13}+384y^{2}zw^{14}+32y^{2}w^{15}-1687500yz^{16}+9421875yz^{15}w-28968750yz^{14}w^{2}+105234375yz^{13}w^{3}-3594131yz^{12}w^{4}-436138296yz^{11}w^{5}+310064586yz^{10}w^{6}+290316575yz^{9}w^{7}-219361915yz^{8}w^{8}-71425173yz^{7}w^{9}+42135202yz^{6}w^{10}+9045763yz^{5}w^{11}-2328460yz^{4}w^{12}-460920yz^{3}w^{13}+17856yz^{2}w^{14}+4336yzw^{15}+64yw^{16}-1687500z^{17}-260015625z^{16}w+894375000z^{15}w^{2}+136218750z^{14}w^{3}-2657345581z^{13}w^{4}+1878808061z^{12}w^{5}+803833218z^{11}w^{6}-1049114732z^{10}w^{7}+272750585z^{9}w^{8}+43961407z^{8}w^{9}-99146392z^{7}w^{10}+22595104z^{6}w^{11}+10487859z^{5}w^{12}-1769750z^{4}w^{13}-433704z^{3}w^{14}+18704z^{2}w^{15}+3952zw^{16}+32w^{17})}$ |
Hi
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Cover information
Click on a modular curve in the diagram to see information about it.
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The following modular covers realize this modular curve as a fiber product over $X(1)$.
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.