curve #729

y² + xy = x³ − 434165894750903680711585028937973332537073036610x + 118919699047216645459738888164202698002052991684183731818689878950684100

a-invariants: [1, 0, 0, -434165894750903680711585028937973332537073036610, 118919699047216645459738888164202698002052991684183731818689878950684100]
rank (lower bound): ≥ 28
torsion subgroup: trivial
conductor (N): 422367126030526227285230492207668179902850039758472657342841281824734303927597787619945708073765087852748052210
discriminant (Δ): -871520537145193129691926453931032617739180312054259608942737153964394481608950798682903179131341681988791188221357432544511212449823930675200000
Faltings height: 26.3609
naive height: 340.8368
primes of bad reduction: 2, 3, 5, 7, 11, 13, 19, 31, 43, 61, 83, 113, 4860108016055346764861, 28531243055359640531929213801903207653957441924525724511352289127381924357
regulator: 1160426835203713021066312428485.768481441288118869683311696484326861107734813729809652522101236
submitted by: wgxli
submitted at: 2026-09-14 01:50:59 UTC
last updated: 2026-09-14 01:50:59 UTC

Witness: 28 independent points log in to add more points →

(103644624081864219075172980/169, -634160445292155063395220375312430297770/2197)
(8661123594744228299579295394845186990937777667020/28028516880803780746779201, 17722593668414565620373121067551796395938914339197365963052756270809507890/148388476765178775304033820158933627551)
(19705852102432398033999190/49, 33180643937075812726900343759051320120/343)
(1386228402949984018636939948606969094220/2130248670416689, -32897213184366653691567962796223719349381705029780889043010/98320811497841511203113)
(42777648336794027573589220/81, 140085630157579879311646946801719038910/729)
(69399971181912630464644791220/3481, 18272790544372588156604763170988514067436410/205379)
(27836119209662076568101940/9, 143793502771397880340771173682876819930/27)
(3311198543176391105347180/9, -2559334330343205937233174768755313470/27)
(-22227967159179420802757875143659258920254438050/35882541187075701571489, 2633906590713808379295914585145526008742546251252259712680788792168830/6797117664471941390191673157909263)
(-175541878043321103371259740/961, 13058248720565965388849247309538128388690/29791)
(39271962203713985039719849823071694058878159950/139636663433206204246921, -7203604397773469771866707166783154567288884649920085180072910171208810/52179413637570999386988692346080219)
(1147575592393833540115937412/361, -38104333224304864062657897046655388579462/6859)
(155157254286918850806378173510323875730/274913079733009, -1055865835807737466180699043759461148842066359858473188370/4558197143252394923527)
(21595224000858750393183011665511620/583923694201, -143143098773440263187317784858065485957893447584002090/446204706999999949)
(-585073210943044040349140, -415525853490464888158434454153886450)
(3775368620438261863557795/4, 5932668046171621549375242294362178195/8)
(1434317103915691450813476424572520/9101350801, -202537425429376543947527139065401355563189114347110/868277967766201)
(-23148438490405601905984720506385635998990420/45586914685278912009, 140528100549494472398706144532151080106319828655565345173679288830/307794106800873055709227010277)
(916097988058774822357946274250562389143125380/1850480042760721211449, 12664779465020354304828797398348970868274420373256141350549385398870/79602481858948128327048663981293)
(18599785828470284030317683563601830451326116/15621629429312697025, 70123476379022621833074078753994616298352851955991237736368588974/61743251602252837641001902625)
(1747171185064885847993527126242817775872290055150/3547864557926750497877281, 1046844646683771494005550437879112987509154813202078247315329316544601410/6682678192856483493148782504751002929)
(11983820084081517679325235122413333814340040/28455890968225757569, 15751405682597900322001075063699732495956685879608091782670426250/151795285105884546822699314753)
(-1578903194992885206974485823535932/2648852089, 55532589344073624998061451471208672386255734281066/136328470464563)
(-3276722091364540809870543872255432450350556/7237915567969128025, 9189020339560257374243574762685888991464647248025681074513856982/19472435276255787729656396125)
(-9165038325341462635091285679553563037444108397696/25418727939297695224236169, 61271427399634497582837891583740868516549276951771377630468833891591241286/128153573046488204471523609688905395803)
(69469635347859184752564837668960170321363238565/41726636896378915743376, 17070087225793273664343309859862386447603737035440723622857703696482995/8523541381293108662496100166685376)
(-2984113776152777120999871430448677772/6739875400129, 8287916351203130408098398471092519899651489667651351138/17497572502910700383)
(2027681435134777551427245202515367300/3353341051369, 1709260342301875376187189831226124875308371322357566470/6140681726700580597)

Commentary

Specialization at T = 1730/1943 of the rank-17 fibration of X1092 given by: y^2 + (Lx+B)y = x(x+D)(x+E), where B = pE + qD + pq(L-p-q), L = 109336T^2 + 22126T + 852111, p = 190386T^2 + 63117T - 208818, q = 750092T^2 - 1506004T + 80940, D = 54(4911166512T^4 - 3823797642T^3 - 29521574359T^2 + 15680855866T + 6154886748), E = 5(124438602862T^4 + 358460463503T^3 - 278881955603T^2 - 2983058352T + 5265641160).

last edited by wgxli at 2026-09-17 17:25:41 UTC · history