Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ x^{2} - x y + 3 x z + x w - y^{2} + y z - 3 y w + z^{2} - z w - w^{2} $ |
| $=$ | $2 x^{2} y - 2 x y^{2} + 3 x y z - x y w - y^{2} z + y z^{2} - 2 y z w - z^{2} w + z w^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - x^{4} y^{2} - 7 x^{4} y z - x^{4} z^{2} - x^{3} y^{3} + 26 x^{3} y^{2} z - 15 x^{3} y z^{2} + \cdots + z^{6} $ |
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This modular curve has 1 rational CM point but no rational cusps or other known 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 150 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2^6\cdot5^5\,\frac{54xy^{2}z^{22}-2592xy^{2}z^{21}w+133974xy^{2}z^{20}w^{2}+36908808160xy^{2}z^{19}w^{3}+888062615810xy^{2}z^{18}w^{4}+7697368812096xy^{2}z^{17}w^{5}+45974762939002xy^{2}z^{16}w^{6}+183598817024336xy^{2}z^{15}w^{7}+537834722965080xy^{2}z^{14}w^{8}+1157930747340480xy^{2}z^{13}w^{9}+1848334646255904xy^{2}z^{12}w^{10}+2165987503454688xy^{2}z^{11}w^{11}+1848334646255904xy^{2}z^{10}w^{12}+1157930747340480xy^{2}z^{9}w^{13}+537834722965080xy^{2}z^{8}w^{14}+183598817024336xy^{2}z^{7}w^{15}+45974762939002xy^{2}z^{6}w^{16}+7697368812096xy^{2}z^{5}w^{17}+888062615810xy^{2}z^{4}w^{18}+36908808160xy^{2}z^{3}w^{19}+133974xy^{2}z^{2}w^{20}-2592xy^{2}zw^{21}+54xy^{2}w^{22}-81xyz^{23}+3753xyz^{22}w-282501xyz^{21}w^{2}-127236676659xyz^{20}w^{3}-2474262158775xyz^{19}w^{4}-22117420083865xyz^{18}w^{5}-124493574330323xyz^{17}w^{6}-479600360749509xyz^{16}w^{7}-1323668856853562xyz^{15}w^{8}-2644075293789530xyz^{14}w^{9}-3774693108743054xyz^{13}w^{10}-3711163395063502xyz^{12}w^{11}-2310319829502386xyz^{11}w^{12}-734704981651522xyz^{10}w^{13}+100427088331190xyz^{9}w^{14}+241518546869438xyz^{8}w^{15}+135239990804502xyz^{7}w^{16}+43985376703676xyz^{6}w^{17}+9239272942006xyz^{5}w^{18}+1086805022820xyz^{4}w^{19}+71873263458xyz^{3}w^{20}+85428xyz^{2}w^{21}+54xyzw^{22}-1890xz^{22}w^{2}+148109395148xz^{21}w^{3}+3138205486532xz^{20}w^{4}+32008825903282xz^{19}w^{5}+206388234536468xz^{18}w^{6}+935757355947680xz^{17}w^{7}+3135173127279214xz^{16}w^{8}+7965303502118196xz^{15}w^{9}+15511510402087446xz^{14}w^{10}+23196250087455534xz^{13}w^{11}+26552124749264140xz^{12}w^{12}+23196250087455534xz^{11}w^{13}+15511510402087446xz^{10}w^{14}+7965303502118196xz^{9}w^{15}+3135173127279214xz^{8}w^{16}+935757355947680xz^{7}w^{17}+206388234536468xz^{6}w^{18}+32008825903282xz^{5}w^{19}+3138205486532xz^{4}w^{20}+148109395148xz^{3}w^{21}-1890xz^{2}w^{22}-27y^{2}z^{23}+1296y^{2}z^{22}w+22653y^{2}z^{21}w^{2}-51472351268y^{2}z^{20}w^{3}-1010702293505y^{2}z^{19}w^{4}-10559626418800y^{2}z^{18}w^{5}-67299231158881y^{2}z^{17}w^{6}-301815649478368y^{2}z^{16}w^{7}-990245123469314y^{2}z^{15}w^{8}-2438118544010700y^{2}z^{14}w^{9}-4536375098894938y^{2}z^{13}w^{10}-6360435948478404y^{2}z^{12}w^{11}-6663791659760482y^{2}z^{11}w^{12}-5193364288796424y^{2}z^{10}w^{13}-3035249704601810y^{2}z^{9}w^{14}-1330043942904884y^{2}z^{8}w^{15}-437205757648706y^{2}z^{7}w^{16}-103691964017248y^{2}z^{6}w^{17}-17726340186378y^{2}z^{5}w^{18}-1727293624740y^{2}z^{4}w^{19}-104891478594y^{2}z^{3}w^{20}+9396y^{2}z^{2}w^{21}-162y^{2}zw^{22}-27yz^{24}+1674yz^{23}w-174501yz^{22}w^{2}+41273573088yz^{21}w^{3}+717624825717yz^{20}w^{4}+6060312108162yz^{19}w^{5}+28200855228139yz^{18}w^{6}+72057497428826yz^{17}w^{7}+17166169262772yz^{16}w^{8}-639805909929634yz^{15}w^{9}-2853872290094362yz^{14}w^{10}-7163321394096682yz^{13}w^{11}-12113556930287536yz^{12}w^{12}-14382376849894852yz^{11}w^{13}-12208838642173244yz^{10}w^{14}-7579772690296218yz^{9}w^{15}-3481560451150460yz^{8}w^{16}-1183171507177404yz^{7}w^{17}-291434512379968yz^{6}w^{18}-50033381528518yz^{5}w^{19}-5314793245542yz^{4}w^{20}-279710552938yz^{3}w^{21}+24084yz^{2}w^{22}-432yzw^{23}-27z^{24}w-3699z^{23}w^{2}+56573030985z^{22}w^{3}+1118554776747z^{21}w^{4}+10497069468989z^{20}w^{5}+60335009537065z^{19}w^{6}+234511672498863z^{18}w^{7}+626142020435163z^{17}w^{8}+1092873508349566z^{16}w^{9}+915633231971492z^{15}w^{10}-932282097204908z^{14}w^{11}-4444459089586622z^{13}w^{12}-7641507322795836z^{12}w^{13}-8271819833115284z^{11}w^{14}-6320525655810058z^{10}w^{15}-3560138913519044z^{9}w^{16}-1504461723853704z^{8}w^{17}-475718751848896z^{7}w^{18}-110639745997978z^{6}w^{19}-17947974043936z^{5}w^{20}-1853086977544z^{4}w^{21}-91536297770z^{3}w^{22}-3132z^{2}w^{23}}{70145546346xy^{2}z^{22}+5072792284192xy^{2}z^{21}w+41883226352176xy^{2}z^{20}w^{2}+325351746495040xy^{2}z^{19}w^{3}+1573916174997690xy^{2}z^{18}w^{4}+6038480704564064xy^{2}z^{17}w^{5}+17960404177577918xy^{2}z^{16}w^{6}+43034206881428224xy^{2}z^{15}w^{7}+83911289773751520xy^{2}z^{14}w^{8}+134449905434497920xy^{2}z^{13}w^{9}+178045416809737516xy^{2}z^{12}w^{10}+195452501233949952xy^{2}z^{11}w^{11}+178045416809737516xy^{2}z^{10}w^{12}+134449905434497920xy^{2}z^{9}w^{13}+83911289773751520xy^{2}z^{8}w^{14}+43034206881428224xy^{2}z^{7}w^{15}+17960404177577918xy^{2}z^{6}w^{16}+6038480704564064xy^{2}z^{5}w^{17}+1573916174997690xy^{2}z^{4}w^{18}+325351746495040xy^{2}z^{3}w^{19}+41883226352176xy^{2}z^{2}w^{20}+5072792284192xy^{2}zw^{21}+70145546346xy^{2}w^{22}-507577902239xyz^{23}-12490058063673xyz^{22}w-128231875045179xyz^{21}w^{2}-880693308887715xyz^{20}w^{3}-4251348692338910xyz^{19}w^{4}-15569762121003976xyz^{18}w^{5}-44481906283717117xyz^{17}w^{6}-101233503010807571xyz^{16}w^{7}-185583392295620150xyz^{15}w^{8}-275403505042598630xyz^{14}w^{9}-330586951479361844xyz^{13}w^{10}-318529571583555088xyz^{12}w^{11}-241717305481976114xyz^{11}w^{12}-138156031886991310xyz^{10}w^{13}-52138287769775530xyz^{9}w^{14}-4834852687149466xyz^{8}w^{15}+9741586422298358xyz^{7}w^{16}+8483578960504144xyz^{6}w^{17}+4151166801661345xyz^{5}w^{18}+1402446810099815xyz^{4}w^{19}+329840849616891xyz^{3}w^{20}+57797847090627xyz^{2}w^{21}+4775651317866xyzw^{22}+402359582720xyw^{23}+629355315200xz^{24}+15279891415040xz^{23}w+180737088989570xz^{22}w^{2}+1371059651998690xz^{21}w^{3}+7512408194403500xz^{20}w^{4}+31440007680901850xz^{19}w^{5}+104287129211358060xz^{18}w^{6}+280304842723607530xz^{17}w^{7}+619916511505865360xz^{16}w^{8}+1139537258398854700xz^{15}w^{9}+1752912394347344100xz^{14}w^{10}+2266176915671878440xz^{13}w^{11}+2468092482380135920xz^{12}w^{12}+2266176915671878440xz^{11}w^{13}+1752912394347344100xz^{10}w^{14}+1139537258398854700xz^{9}w^{15}+619916511505865360xz^{8}w^{16}+280304842723607530xz^{7}w^{17}+104287129211358060xz^{6}w^{18}+31440007680901850xz^{5}w^{19}+7512408194403500xz^{4}w^{20}+1371059651998690xz^{3}w^{21}+180737088989570xz^{2}w^{22}+15279891415040xzw^{23}+629355315200xw^{24}-262068505653y^{2}z^{23}-4609598666256y^{2}z^{22}w-60618417479433y^{2}z^{21}w^{2}-446060114091320y^{2}z^{20}w^{3}-2440746134110270y^{2}z^{19}w^{4}-10074601688252052y^{2}z^{18}w^{5}-32839105740662699y^{2}z^{17}w^{6}-86266010675564192y^{2}z^{16}w^{7}-185359617616909610y^{2}z^{15}w^{8}-328952776322404360y^{2}z^{14}w^{9}-485151741356247188y^{2}z^{13}w^{10}-596834433310572736y^{2}z^{12}w^{11}-613481710831096678y^{2}z^{11}w^{12}-526872811933382960y^{2}z^{10}w^{13}-377412115382882910y^{2}z^{9}w^{14}-224637533805740832y^{2}z^{8}w^{15}-110411302012304274y^{2}z^{7}w^{16}-44419444021506012y^{2}z^{6}w^{17}-14354360487626745y^{2}z^{5}w^{18}-3706938349701320y^{2}z^{4}w^{19}-696991543165043y^{2}z^{3}w^{20}-107620235962356y^{2}z^{2}w^{21}-6989144934718y^{2}zw^{22}-629355315200y^{2}w^{23}+191922959307yz^{24}+3407175939286yz^{23}w+36982533776331yz^{22}w^{2}+214351884653722yz^{21}w^{3}+840949245321860yz^{20}w^{4}+1886962249676218yz^{19}w^{5}+328392815720979yz^{18}w^{6}-16708413270391786yz^{17}w^{7}-79680070710736902yz^{16}w^{8}-232164817256568320yz^{15}w^{9}-500037210781471688yz^{14}w^{10}-845132879903782904yz^{13}w^{11}-1151128753936844094yz^{12}w^{12}-1280759107355587788yz^{11}w^{13}-1171968858079773870yz^{10}w^{14}-884259751895481052yz^{9}w^{15}-549655242394978476yz^{8}w^{16}-280333059651160006yz^{7}w^{17}-116354220099713107yz^{6}w^{18}-38769578527926590yz^{5}w^{19}-10171038261354953yz^{4}w^{20}-2015057620142004yz^{3}w^{21}-291613286168404yz^{2}w^{22}-25596755231568yzw^{23}-1258710630400yw^{24}+240392339456z^{25}+5469112366027z^{24}w+60854090954659z^{23}w^{2}+419766323625287z^{22}w^{3}+2064786179631315z^{21}w^{4}+7492587841838010z^{20}w^{5}+20598875645858468z^{19}w^{6}+42456325280849811z^{18}w^{7}+61641019050167863z^{17}w^{8}+46176775082341230z^{16}w^{9}-46392382825401750z^{15}w^{10}-233461981619998208z^{14}w^{11}-471682348938591696z^{13}w^{12}-664894373434742958z^{12}w^{13}-724928747256912750z^{11}w^{14}-634498210943517950z^{10}w^{15}-453005430364486462z^{9}w^{16}-265433748380164814z^{8}w^{17}-127600426494892732z^{7}w^{18}-50015963619814555z^{6}w^{19}-15810729430409015z^{5}w^{20}-3934828102702063z^{4}w^{21}-752151022349841z^{3}w^{22}-102101783250948z^{2}w^{23}-9146350960640zw^{24}-388962975744w^{25}}$ 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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.
