Weierstrass model Weierstrass model
$ y^{2} $ | $=$ | $ x^{3} - 156x + 560 $ |
This modular curve has infinitely many rational points but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.
Maps to other modular curves
$j$-invariant map
of degree 36 from the Weierstrass model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -2^6\cdot3^3\,\frac{72x^{2}y^{22}-20736x^{2}y^{21}z+66071376x^{2}y^{20}z^{2}-7790059008x^{2}y^{19}z^{3}+4375336974336x^{2}y^{18}z^{4}+4985740960727040x^{2}y^{17}z^{5}-501074378134124544x^{2}y^{16}z^{6}-277483490206595481600x^{2}y^{15}z^{7}-5861699065625745162240x^{2}y^{14}z^{8}+4324709756738039722278912x^{2}y^{13}z^{9}+329667428914960828019834880x^{2}y^{12}z^{10}-17163761524183814188119883776x^{2}y^{11}z^{11}-2823625506027904125853413408768x^{2}y^{10}z^{12}-50028342581779656597677832929280x^{2}y^{9}z^{13}+7569962252869129535767158554886144x^{2}y^{8}z^{14}+410138981317838374808185122966208512x^{2}y^{7}z^{15}-1398428803009386959701186751194202112x^{2}y^{6}z^{16}-675693506531608198831956718243175989248x^{2}y^{5}z^{17}-17455965195897634218455152267197451075584x^{2}y^{4}z^{18}+138279524246449313066994033512789733015552x^{2}y^{3}z^{19}+14424209932890985979015033427455750499729408x^{2}y^{2}z^{20}+262891875039988366518862532323147257936347136x^{2}yz^{21}+1652463188908291784789287575739126161761894400x^{2}z^{22}-12456xy^{22}z+7910784xy^{21}z^{2}-4158190080xy^{20}z^{3}+1270797465600xy^{19}z^{4}-353135151180288xy^{18}z^{5}-228367651713196032xy^{17}z^{6}+10987345214992834560xy^{16}z^{7}+8393692393242846167040xy^{15}z^{8}+331669815640822765191168xy^{14}z^{9}-93667847409979964371501056xy^{13}z^{10}-8404179786303771040563068928xy^{12}z^{11}+227309558556696266611712065536xy^{11}z^{12}+54272258865244967699374135050240xy^{10}z^{13}+1352863998269545597862635106205696xy^{9}z^{14}-115166339723902528044386764537724928xy^{8}z^{15}-7179680725623164795520466026026237952xy^{7}z^{16}-12970880885073381628638137933862273024xy^{6}z^{17}+9916811637539905651032384851899915960320xy^{5}z^{18}+278634375207428652637109291884612077748224xy^{4}z^{19}-1327361859835305048714231784573709158711296xy^{3}z^{20}-198113757951957211631596038542491991446388736xy^{2}z^{21}-3680486103988746283516811527231427812987502592xyz^{22}-23134484338280755032110389935977217594067255296xz^{23}+y^{24}-864y^{23}z+91008y^{22}z^{2}-300423168y^{21}z^{3}+56977219584y^{20}z^{4}-89646064146432y^{19}z^{5}+15872654817718272y^{18}z^{6}+8228758593635942400y^{17}z^{7}-39281352687039209472y^{16}z^{8}-186949508101111824777216y^{15}z^{9}-11490512322517848167546880y^{14}z^{10}+1194573875317106180345561088y^{13}z^{11}+145703443569522123784489795584y^{12}z^{12}+602273056310855008821669003264y^{11}z^{13}-556130439395782124535185028415488y^{10}z^{14}-23223346653160281595907977978576896y^{9}z^{15}+503609929045300572741134851982229504y^{8}z^{16}+58043329628941459334341835265530658816y^{7}z^{17}+936810916181580890766797003697439113216y^{6}z^{18}-38416525411956664712465062149718907289600y^{5}z^{19}-1648392741225990506934000105682867668910080y^{4}z^{20}-11508127763091235441529608310045732014915584y^{3}z^{21}+469864165298032499052641491148041039852339200y^{2}z^{22}+10515674415315171269765445591755355124968456192yz^{23}+66098526334586371070997888275252670369437319168z^{24}}{360x^{2}y^{22}-429848208x^{2}y^{20}z^{2}+33394315668480x^{2}y^{18}z^{4}-641620123326609408x^{2}y^{16}z^{6}+5446652204005988106240x^{2}y^{14}z^{8}-25616596484031736736710656x^{2}y^{12}z^{10}+73961309738734754217597075456x^{2}y^{10}z^{12}-137112146498700920819737750929408x^{2}y^{8}z^{14}+164356144773294938755505905496752128x^{2}y^{6}z^{16}-123502816695595965427738399662743224320x^{2}y^{4}z^{18}+53007286014128058324741977067696829759488x^{2}y^{2}z^{20}-9932160377396765900299707927935936643465216x^{2}z^{22}-59688xy^{22}z+23124413952xy^{20}z^{3}-1056989767104000xy^{18}z^{5}+15682812865550622720xy^{16}z^{7}-114037872088719122104320xy^{14}z^{9}+482466125175646794772119552xy^{12}z^{11}-1287892604701596880335028617216xy^{10}z^{13}+2245603257763282916242332183429120xy^{8}z^{15}-2561301670728696304854795045826461696xy^{6}z^{17}+1846567274515633524571781715670163521536xy^{4}z^{19}-765093106102136948603478889139792177528832xy^{2}z^{21}+139050244103719756361340854172155410397528064xz^{23}-y^{24}+6053760y^{22}z^{2}-937978921728y^{20}z^{4}+25373708134133760y^{18}z^{6}-266437505592623112192y^{16}z^{8}+1477708389477878926934016y^{14}z^{10}-4949563541404425143218864128y^{12}z^{12}+10667585497097166800538681999360y^{10}z^{14}-15159500378579506870733001782722560y^{8}z^{16}+14123163413259123109322426086909280256y^{6}z^{18}-8284113517212665488930401946575656976384y^{4}z^{20}+2764042598891615782781031674293659063287808y^{2}z^{22}-397286414372550270225497833011465236653277184z^{24}}$ |
Hi
|
Cover information
Click on a modular curve in the diagram to see information about it.
|
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.