Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ x^{2} + 4 x y + y^{2} - z^{2} + z w - w^{2} $ |
| $=$ | $x^{3} + 2 x^{2} y - x y^{2} + 4 x z^{2} - 4 x z w + 4 x w^{2} + y^{3} + 3 y z^{2} - 3 y z w + \cdots - 2 w^{3}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 7 x^{6} + 54 x^{5} y + 129 x^{4} y^{2} + 9 x^{4} z^{2} + 106 x^{3} y^{3} + 72 x^{3} y z^{2} + \cdots + 9 z^{6} $ |
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This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from canonical model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Maps to other modular curves
$j$-invariant map
of degree 162 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 3^3\,\frac{669440xyz^{25}-8368000xyz^{24}w+52739388xyz^{23}w^{2}-221574962xyz^{22}w^{3}+693457414xyz^{21}w^{4}-1714612410xyz^{20}w^{5}+3467204030xyz^{19}w^{6}-5853089536xyz^{18}w^{7}+8326233324xyz^{17}w^{8}-9948561908xyz^{16}w^{9}+9738122080xyz^{15}w^{10}-7187580270xyz^{14}w^{11}+2655026690xyz^{13}w^{12}+2655026690xyz^{12}w^{13}-7187580270xyz^{11}w^{14}+9738122080xyz^{10}w^{15}-9948561908xyz^{9}w^{16}+8326233324xyz^{8}w^{17}-5853089536xyz^{7}w^{18}+3467204030xyz^{6}w^{19}-1714612410xyz^{5}w^{20}+693457414xyz^{4}w^{21}-221574962xyz^{3}w^{22}+52739388xyz^{2}w^{23}-8368000xyzw^{24}+669440xyw^{25}+949888xz^{26}-12348544xz^{25}w+82904482xz^{24}w^{2}-377426584xz^{23}w^{3}+1305027643xz^{22}w^{4}-3647452501xz^{21}w^{5}+8570326294xz^{20}w^{6}-17376478249xz^{19}w^{7}+30944951560xz^{18}w^{8}-48992795809xz^{17}w^{9}+69532223959xz^{16}w^{10}-88960269526xz^{15}w^{11}+102986644516xz^{14}w^{12}-108111564370xz^{13}w^{13}+102986644516xz^{12}w^{14}-88960269526xz^{11}w^{15}+69532223959xz^{10}w^{16}-48992795809xz^{9}w^{17}+30944951560xz^{8}w^{18}-17376478249xz^{7}w^{19}+8570326294xz^{6}w^{20}-3647452501xz^{5}w^{21}+1305027643xz^{4}w^{22}-377426584xz^{3}w^{23}+82904482xz^{2}w^{24}-12348544xzw^{25}+949888xw^{26}+261120y^{3}z^{24}-3133440y^{3}z^{23}w+19579302y^{3}z^{22}w^{2}-83245602y^{3}z^{21}w^{3}+269535609y^{3}z^{20}w^{4}-706647060y^{3}z^{19}w^{5}+1558309917y^{3}z^{18}w^{6}-2961622602y^{3}z^{17}w^{7}+4928870169y^{3}z^{16}w^{8}-7258648080y^{3}z^{15}w^{9}+9526299660y^{3}z^{14}w^{10}-11193599718y^{3}z^{13}w^{11}+11808342570y^{3}z^{12}w^{12}-11193599718y^{3}z^{11}w^{13}+9526299660y^{3}z^{10}w^{14}-7258648080y^{3}z^{9}w^{15}+4928870169y^{3}z^{8}w^{16}-2961622602y^{3}z^{7}w^{17}+1558309917y^{3}z^{6}w^{18}-706647060y^{3}z^{5}w^{19}+269535609y^{3}z^{4}w^{20}-83245602y^{3}z^{3}w^{21}+19579302y^{3}z^{2}w^{22}-3133440y^{3}zw^{23}+261120y^{3}w^{24}+156928y^{2}z^{25}-1961600y^{2}z^{24}w+12361392y^{2}z^{23}w^{2}-51922408y^{2}z^{22}w^{3}+162445064y^{2}z^{21}w^{4}-401459184y^{2}z^{20}w^{5}+811304104y^{2}z^{19}w^{6}-1368654512y^{2}z^{18}w^{7}+1945776168y^{2}z^{17}w^{8}-2323904896y^{2}z^{16}w^{9}+2274252128y^{2}z^{15}w^{10}-1678496616y^{2}z^{14}w^{11}+620024968y^{2}z^{13}w^{12}+620024968y^{2}z^{12}w^{13}-1678496616y^{2}z^{11}w^{14}+2274252128y^{2}z^{10}w^{15}-2323904896y^{2}z^{9}w^{16}+1945776168y^{2}z^{8}w^{17}-1368654512y^{2}z^{7}w^{18}+811304104y^{2}z^{6}w^{19}-401459184y^{2}z^{5}w^{20}+162445064y^{2}z^{4}w^{21}-51922408y^{2}z^{3}w^{22}+12361392y^{2}z^{2}w^{23}-1961600y^{2}zw^{24}+156928y^{2}w^{25}+513408yz^{26}-6674304yz^{25}w+44990454yz^{24}w^{2}-206170248yz^{23}w^{3}+718768227yz^{22}w^{4}-2027269893yz^{21}w^{5}+4807652646yz^{20}w^{6}-9833349873yz^{19}w^{7}+17649112548yz^{18}w^{8}-28127039097yz^{17}w^{9}+40127214027yz^{16}w^{10}-51532746582yz^{15}w^{11}+59793992796yz^{14}w^{12}-62817474810yz^{13}w^{13}+59793992796yz^{12}w^{14}-51532746582yz^{11}w^{15}+40127214027yz^{10}w^{16}-28127039097yz^{9}w^{17}+17649112548yz^{8}w^{18}-9833349873yz^{7}w^{19}+4807652646yz^{6}w^{20}-2027269893yz^{5}w^{21}+718768227yz^{4}w^{22}-206170248yz^{3}w^{23}+44990454yz^{2}w^{24}-6674304yzw^{25}+513408yw^{26}-588800z^{27}+7948800z^{26}w-54036840z^{25}w^{2}+244900500z^{24}w^{3}-828917094z^{23}w^{4}+2228275581z^{22}w^{5}-4937686635z^{21}w^{6}+9228184899z^{20}w^{7}-14728682421z^{19}w^{8}+20140405365z^{18}w^{9}-23407561935z^{17}w^{10}+22481179032z^{16}w^{11}-16392671646z^{15}w^{12}+6019545594z^{14}w^{13}+6019545594z^{13}w^{14}-16392671646z^{12}w^{15}+22481179032z^{11}w^{16}-23407561935z^{10}w^{17}+20140405365z^{9}w^{18}-14728682421z^{8}w^{19}+9228184899z^{7}w^{20}-4937686635z^{6}w^{21}+2228275581z^{5}w^{22}-828917094z^{4}w^{23}+244900500z^{3}w^{24}-54036840z^{2}w^{25}+7948800zw^{26}-588800w^{27}}{10460xyz^{25}-130750xyz^{24}w+977796xyz^{23}w^{2}-5230154xyz^{22}w^{3}+21145390xyz^{21}w^{4}-66977316xyz^{20}w^{5}+170131466xyz^{19}w^{6}-351576304xyz^{18}w^{7}+594202716xyz^{17}w^{8}-816423062xyz^{16}w^{9}+887998000xyz^{15}w^{10}-703188774xyz^{14}w^{11}+269055302xyz^{13}w^{12}+269055302xyz^{12}w^{13}-703188774xyz^{11}w^{14}+887998000xyz^{10}w^{15}-816423062xyz^{9}w^{16}+594202716xyz^{8}w^{17}-351576304xyz^{7}w^{18}+170131466xyz^{6}w^{19}-66977316xyz^{5}w^{20}+21145390xyz^{4}w^{21}-5230154xyz^{3}w^{22}+977796xyz^{2}w^{23}-130750xyzw^{24}+10460xyw^{25}+14842xz^{26}-192946xz^{25}w+1239004xz^{24}w^{2}-5220748xz^{23}w^{3}+15645820xz^{22}w^{4}-33321376xz^{21}w^{5}+43847704xz^{20}w^{6}+369302xz^{19}w^{7}-185269823xz^{18}w^{8}+601170941xz^{17}w^{9}-1266281243xz^{16}w^{10}+2055640196xz^{15}w^{11}-2712082232xz^{14}w^{12}+2968895960xz^{13}w^{13}-2712082232xz^{12}w^{14}+2055640196xz^{11}w^{15}-1266281243xz^{10}w^{16}+601170941xz^{9}w^{17}-185269823xz^{8}w^{18}+369302xz^{7}w^{19}+43847704xz^{6}w^{20}-33321376xz^{5}w^{21}+15645820xz^{4}w^{22}-5220748xz^{3}w^{23}+1239004xz^{2}w^{24}-192946xzw^{25}+14842xw^{26}+4080y^{3}z^{24}-48960y^{3}z^{23}w+282348y^{3}z^{22}w^{2}-1041348y^{3}z^{21}w^{3}+2597112y^{3}z^{20}w^{4}-3975300y^{3}z^{19}w^{5}+790536y^{3}z^{18}w^{6}+16107660y^{3}z^{17}w^{7}-57676455y^{3}z^{16}w^{8}+127925088y^{3}z^{15}w^{9}-214391628y^{3}z^{14}w^{10}+287951832y^{3}z^{13}w^{11}-317045850y^{3}z^{12}w^{12}+287951832y^{3}z^{11}w^{13}-214391628y^{3}z^{10}w^{14}+127925088y^{3}z^{9}w^{15}-57676455y^{3}z^{8}w^{16}+16107660y^{3}z^{7}w^{17}+790536y^{3}z^{6}w^{18}-3975300y^{3}z^{5}w^{19}+2597112y^{3}z^{4}w^{20}-1041348y^{3}z^{3}w^{21}+282348y^{3}z^{2}w^{22}-48960y^{3}zw^{23}+4080y^{3}w^{24}+2452y^{2}z^{25}-30650y^{2}z^{24}w+230508y^{2}z^{23}w^{2}-1240942y^{2}z^{22}w^{3}+5044658y^{2}z^{21}w^{4}-16048872y^{2}z^{20}w^{5}+40907182y^{2}z^{19}w^{6}-84763940y^{2}z^{18}w^{7}+143562780y^{2}z^{17}w^{8}-197568730y^{2}z^{16}w^{9}+215138312y^{2}z^{15}w^{10}-170492238y^{2}z^{14}w^{11}+65258254y^{2}z^{13}w^{12}+65258254y^{2}z^{12}w^{13}-170492238y^{2}z^{11}w^{14}+215138312y^{2}z^{10}w^{15}-197568730y^{2}z^{9}w^{16}+143562780y^{2}z^{8}w^{17}-84763940y^{2}z^{7}w^{18}+40907182y^{2}z^{6}w^{19}-16048872y^{2}z^{5}w^{20}+5044658y^{2}z^{4}w^{21}-1240942y^{2}z^{3}w^{22}+230508y^{2}z^{2}w^{23}-30650y^{2}zw^{24}+2452y^{2}w^{25}+8022yz^{26}-104286yz^{25}w+664554yz^{24}w^{2}-2760348yz^{23}w^{3}+8082948yz^{22}w^{4}-16491684yz^{21}w^{5}+19096158yz^{20}w^{6}+11223942yz^{19}w^{7}-121765839yz^{18}w^{8}+360644649yz^{17}w^{9}-735248121yz^{16}w^{10}+1174876116yz^{15}w^{11}-1538140380yz^{14}w^{12}+1679836560yz^{13}w^{13}-1538140380yz^{12}w^{14}+1174876116yz^{11}w^{15}-735248121yz^{10}w^{16}+360644649yz^{9}w^{17}-121765839yz^{8}w^{18}+11223942yz^{7}w^{19}+19096158yz^{6}w^{20}-16491684yz^{5}w^{21}+8082948yz^{4}w^{22}-2760348yz^{3}w^{23}+664554yz^{2}w^{24}-104286yzw^{25}+8022yw^{26}-9200z^{27}+124200z^{26}w-858288z^{25}w^{2}+4001100z^{24}w^{3}-13929816z^{23}w^{4}+38035284z^{22}w^{5}-83688708z^{21}w^{6}+150739620z^{20}w^{7}-224188434z^{19}w^{8}+276265605z^{18}w^{9}-281529279z^{17}w^{10}+234162318z^{16}w^{11}-150040968z^{15}w^{12}+50921166z^{14}w^{13}+50921166z^{13}w^{14}-150040968z^{12}w^{15}+234162318z^{11}w^{16}-281529279z^{10}w^{17}+276265605z^{9}w^{18}-224188434z^{8}w^{19}+150739620z^{7}w^{20}-83688708z^{6}w^{21}+38035284z^{5}w^{22}-13929816z^{4}w^{23}+4001100z^{3}w^{24}-858288z^{2}w^{25}+124200zw^{26}-9200w^{27}}$ 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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.
