Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 3 x^{2} - y z + y w + z^{2} - w^{2} $ |
| $=$ | $5 y^{2} - 3 y z - 2 y w + 2 z^{2} + 2 z w + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 20 x^{4} - 25 x^{3} z - 51 x^{2} y^{2} + 30 x^{2} z^{2} + 24 x y^{2} z - 10 x z^{3} + 36 y^{4} + \cdots + 5 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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 3^3\,\frac{182072180711yz^{17}+1873015071428yz^{16}w-6422751361819yz^{15}w^{2}-14177978780680yz^{14}w^{3}-14103714655720yz^{13}w^{4}-34788430929118yz^{12}w^{5}-15794197787059yz^{11}w^{6}-20462862756958yz^{10}w^{7}+15031776911435yz^{9}w^{8}+1923052171390yz^{8}w^{9}+9196820219428yz^{7}w^{10}+24480935922034yz^{6}w^{11}+18532397920573yz^{5}w^{12}+20436470026120yz^{4}w^{13}+8513843136205yz^{3}w^{14}+5778789774604yz^{2}w^{15}+83591141722yzw^{16}-282828204296yw^{17}+61455624866z^{18}-1533441159190z^{17}w-3036569327803z^{16}w^{2}+7509987314732z^{15}w^{3}+10845907584620z^{14}w^{4}+36554873358452z^{13}w^{5}+50775195521525z^{12}w^{6}+83852604021194z^{11}w^{7}+85604461508744z^{10}w^{8}+108330388628110z^{9}w^{9}+86094868156723z^{8}w^{10}+61448278887370z^{7}w^{11}+26026410246496z^{6}w^{12}+10537211794666z^{5}w^{13}-489078315230z^{4}w^{14}-3134048461916z^{3}w^{15}-2567813290955z^{2}w^{16}-749429033012zw^{17}+51598268733w^{18}}{(z-w)^{3}(1251145811yz^{14}+7613394551yz^{13}w-15930187714yz^{12}w^{2}-292846028489yz^{11}w^{3}-1283514230894yz^{10}w^{4}-3179143788422yz^{9}w^{5}-5215260630552yz^{8}w^{6}-6057101438997yz^{7}w^{7}-5154231929172yz^{6}w^{8}-3266221050512yz^{5}w^{9}-1544886515264yz^{4}w^{10}-538875002624yz^{3}w^{11}-134035569664yz^{2}w^{12}-21814718464yzw^{13}-1780793344yw^{14}-1283890159z^{15}-17977766827z^{14}w-102099072817z^{13}w^{2}-304110762452z^{12}w^{3}-470827135037z^{11}w^{4}-136140720170z^{10}w^{5}+1032251675154z^{9}w^{6}+2593846258734z^{8}w^{7}+3543834377829z^{7}w^{8}+3308236085524z^{6}w^{9}+2245036104592z^{5}w^{10}+1124613226048z^{4}w^{11}+410621374208z^{3}w^{12}+104475573248z^{2}w^{13}+16660901888zw^{14}+1252442112w^{15})}$ |
Hi
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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.