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
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -3^3\,\frac{50106768389649yz^{17}+543555824738598yz^{16}w-392774916139296yz^{15}w^{2}-914419494134880yz^{14}w^{3}+855640913966145yz^{13}w^{4}+536672322157062yz^{12}w^{5}-881883898875556yz^{11}w^{6}-124117560113528yz^{10}w^{7}+496331936495915yz^{9}w^{8}-114806887863510yz^{8}w^{9}-95410625208648yz^{7}w^{10}+41092269584544yz^{6}w^{11}-4425084219618yz^{5}w^{12}+21410529756420yz^{4}w^{13}-23901598250280yz^{3}w^{14}+11352671632064yz^{2}w^{15}-2670920111927yzw^{16}+272900289614yw^{17}+171872676653294z^{18}-190754283750090z^{17}w-286249553183427z^{16}w^{2}+763151737643512z^{15}w^{3}-71494180530045z^{14}w^{4}-700530951996318z^{13}w^{5}+271650023729075z^{12}w^{6}+502360661151904z^{11}w^{7}-393152645856004z^{10}w^{8}-129689317734990z^{9}w^{9}+177666455766507z^{8}w^{10}+57887023511520z^{7}w^{11}-130671074520386z^{6}w^{12}+46328923556556z^{5}w^{13}+18297283806930z^{4}w^{14}-22954631418256z^{3}w^{15}+9507762876730z^{2}w^{16}-2002487123842zw^{17}+187063769197w^{18}}{(z+w)^{2}(187587100672yz^{15}-2381017251840yz^{14}w+11173111726080yz^{13}w^{2}-28080210165760yz^{12}w^{3}+44640894489600yz^{11}w^{4}-49640982405504yz^{10}w^{5}+40281719355520yz^{9}w^{6}-22686588152640yz^{8}w^{7}+6660153377280yz^{7}w^{8}+1481844634000yz^{6}w^{9}-2633156764899yz^{5}w^{10}+1210013391360yz^{4}w^{11}-203495986720yz^{3}w^{12}-15720264960yz^{2}w^{13}+6257798400yzw^{14}-98766848yw^{15}+107040735232z^{16}-451476455424z^{15}w-189785702400z^{14}w^{2}+4351712215040z^{13}w^{3}-10666219914240z^{12}w^{4}+12367323889536z^{11}w^{5}-5795071847552z^{10}w^{6}-4380368040000z^{9}w^{7}+10620215826480z^{8}w^{8}-10044300592280z^{7}w^{9}+5718570461526z^{6}w^{10}-1917656332782z^{5}w^{11}+231390638900z^{4}w^{12}+73311709440z^{3}w^{13}-26284583040z^{2}w^{14}+1412265472zw^{15}+103044096w^{16})}$ |
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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.