Embedded model Embedded model in $\mathbb{P}^{6}$
| $ 0 $ | $=$ | $ y^{2} v + y u v - z w v - z t v - t^{2} v - u^{2} v $ |
| $=$ | $x y v + x z v - x u v + y^{2} v - z w v - z t v + w t v + w u v + t u v$ |
| $=$ | $x z v + x t v + y t v + z t v - w t v - w u v - t^{2} v - t u v$ |
| $=$ | $2 x y v - x u v + y^{2} v - y u v - z^{2} v + z w v + z t v + z u v - t u v$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 11 x^{7} + 24 x^{6} z + 369 x^{5} y^{2} - 6 x^{5} z^{2} + 1140 x^{4} y^{2} z - 45 x^{4} z^{3} + \cdots - 2 z^{7} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ 15x^{11} + 165x^{6} - 15x $ |
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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 to other modular curves
$j$-invariant map
of degree 120 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{1}{2^9\cdot7\cdot401^9}\cdot\frac{6340434025157377977450110019076518750xtu^{9}+991438632368355949392714832850107500xtu^{7}v^{2}-1507134313911064928073432676389999000xtu^{5}v^{4}+16845557549765974716775751788449600xtu^{3}v^{6}-567046788789316524171862546281600xtuv^{8}+3667848463888146892601809461489459375xu^{10}+1238510499076164014252019678771416250xu^{8}v^{2}-854494446475255120302141686214184500xu^{6}v^{4}-11618376899683290549231398354755200xu^{4}v^{6}-134087231105207298717081683236800xu^{2}v^{8}-12209850831914440136186109939584xv^{10}+9485083693485971607326340172787465625ytu^{9}+1562348808365468033095612586220701250ytu^{7}v^{2}-1623544915167400234368080833877385000ytu^{5}v^{4}+13830506117361711500799334311511200ytu^{3}v^{6}-598812876424160727298849448104320ytuv^{8}-6821640205434845041075116286478400000yu^{10}-961457872716327616350133345984209375yu^{8}v^{2}-223293475551507865918160364394717500yu^{6}v^{4}-3816662023143265804268350075794000yu^{4}v^{6}+96783307997743122502781941485600yu^{2}v^{8}-11067180214931709605334535219072yv^{10}+13470357987410356331738539374176409375ztu^{9}+4211255899767645911200295341361565000ztu^{7}v^{2}-2087160190018303289542642624621626000ztu^{5}v^{4}+26898107627411092697443309708848000ztu^{3}v^{6}-757247906783012674870750416197760ztuv^{8}-2137511554176744547980778761585721875zu^{10}-938203271159794533091757811770741250zu^{8}v^{2}-208551031527629779052519923999533000zu^{6}v^{4}-27082633812804405777278537877718800zu^{4}v^{6}-192497244709498612952491862904000zu^{2}v^{8}+569073022206037312517995667968zv^{10}-16975083545347312547750602230766790625wtu^{9}-2321501409913520263758737576178701250wtu^{7}v^{2}+3602663759237454231629701808705688000wtu^{5}v^{4}-36142396023971117275644870835704000wtu^{3}v^{6}+1550472171232370208757591644785280wtuv^{8}-16752943211543492187685341733381068750wu^{10}-5904252812464146664452860000165510625wu^{8}v^{2}+3414806432627778142604471282182794000wu^{6}v^{4}+2047947913789732643307669832010400wu^{4}v^{6}+735651816869214294327389495604000wu^{2}v^{8}-2739813547508269571614442481664wv^{10}-11736494777233452647906603422381068750t^{2}u^{9}-2322279982238195709922646224520280000t^{2}u^{7}v^{2}+3051636403569336829805180914184517000t^{2}u^{5}v^{4}-37366779919814824276107787142340000t^{2}u^{3}v^{6}+1321314782842354658137332691989120t^{2}uv^{8}-17938649205107683351633043515981068750tu^{10}-4012865674603276332776822431214040000tu^{8}v^{2}+3838809590524934278853805029631847500tu^{6}v^{4}+9700984133222631971664981234135600tu^{4}v^{6}+1310939077309118348683501981280640tu^{2}v^{8}-9109708629349801984538198896256tv^{10}+548157216612712519884234169195340625u^{11}-993345264505428840761746743696425625u^{9}v^{2}+770907958026849659643434948654509500u^{7}v^{4}+2662660430089793616946138138843200u^{5}v^{6}+62230855607551956458822265684000u^{3}v^{8}+304358964730965894433793977728uv^{10}}{v^{10}(y-7z+11w+14t+3u)}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
30.120.5.m.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{3}v$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle y$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 11X^{7}+369X^{5}Y^{2}+24X^{6}Z+1140X^{4}Y^{2}Z-6X^{5}Z^{2}+1410X^{3}Y^{2}Z^{2}-45X^{4}Z^{3}+870X^{2}Y^{2}Z^{3}-65X^{3}Z^{4}+270XY^{2}Z^{4}-33X^{2}Z^{5}+33Y^{2}Z^{5}-7XZ^{6}-2Z^{7} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
30.120.5.m.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -\frac{3}{5}x-\frac{1}{5}y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -\frac{123}{625}x^{5}v-\frac{76}{125}x^{4}yv-\frac{94}{125}x^{3}y^{2}v-\frac{58}{125}x^{2}y^{3}v-\frac{18}{125}xy^{4}v-\frac{11}{625}y^{5}v$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{5}x+\frac{2}{5}y$ |
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.
