curve #3568

y² + xy + y = x³ − x² − 638180169290355097052992091392499240632601637683056738584x + 6324382310279887049362229154802940453688099074849575076795002247923693611456690054939

a-invariants: [1, -1, 1, -638180169290355097052992091392499240632601637683056738584, 6324382310279887049362229154802940453688099074849575076795002247923693611456690054939]
rank (lower bound): ≥ 30
torsion subgroup: trivial
conductor (N): 38580071144496576324336113441832189574928903572481789255878697007711215482657399920957504405466939539886959374009723747659469992253209275920544561024022
discriminant (Δ): -644549358233714513108420196964615931105773101068209310010805450771219291860800566837180550735941633800063530591611815601889143133693240761597283763279239764262246646173696
Faltings height: 31.5636
naive height: 404.0463
primes of bad reduction: 2, 7, 11, 13, 17, 23, 31, 61, 3847, 1322641, 134719639, 38022055110651714131457378527019403897201146536776199312834645483224996657835978696758778370186165185197590791854581578875927
regulator: 5819462973823715686528852853513913904934195000.0727272241943327105012294116944040556666191885703279
submitted by: Matej Jurasić
submitted at: 2026-09-28 18:40:09 UTC
last updated: 2026-09-28 18:40:09 UTC

Witness: 30 independent points log in to add more points →

(73084720756152232728841786189, -18709787975231259379699426517381094790088151)
(20297526447490697471190169689, -1316538202895520836635250393104167860579151)
(14871133820807122728984998527, -350257180017174140294352817565298189488903)
(-1187687415016487334919559947, -2660951993616898617702329399054009689831055)
(2140616675013732168458501358853/9, 3114927810237510123265797163579076398737525715/27)
(4036376720989202427006024499, 1952999933715527365006413859736657792870961)
(276697504314459253551308302461/25, -98176190823377678547865179635759119204017731/125)
(31889777951184219989674858893, -4289932544939028434596635441446092959212535)
(16508963433053492507744618701, -536785891596608245064923071744626898075735)
(345055514193785064327519666133/9, 166904497007470654789434445952491889054168435/27)
(22115954759614292989817750989, 1740021215737343229020132381410166895431049)
(48530452749185145056422083021, 9468493394275847034604794328579912494523785)
(261583672180340344834222372793, 133186019588670105628165896776855525796638165)
(14237906594592319297677800243, 352579627083242850153551117025781473352241)
(1241179567938427457086596451777/64, -570563917393081669470293500713702846777517933/512)
(1510722894041196621109423163183/121, 732354458928460469344321415323789493353404605/1331)
(126268999822236084037168988880361/3481, -1141724760308938166311414827788473981623348531933/205379)
(3785586253582774171842995350493441693334335920309/200266463944385734921, 2855591558579594370479036394637255089721109075579960156749274048704327891/2834081560941080461321019365531)
(-1199776279027928029079528913074229648492041973648003/146476269439126258708609, 185947043024614713850313669573860993494676349557195455605014427097754262392487/56059717612525643097811602020433727)
(349804617838240863664375769915582867164409301621093/467358043425624778263001, 772585189094816501312944482342582263466385134597444993519935375380559641147909/319502736522209209450295305991886749)
(697349294044291328038024327597794485709015292499320837163229/76785658656694088337136059390625, -760541158465285879066808737758684477663310575270703000133970472811846821375862580323792567/672852965609981854953857281266763939022556640625)
(1189114665470549080901738634801630348203857213571964218878781/14971501719837857240725389279409, -1237952566690174025569960494673579555871045780587058961168769337340986690866112410058751697/57929268807610270048143179598956642461570166423)
(47674985192236437023855374730676248754832797329489825452151901/838576523625215396887482060274609, 301155762100420100253726610066242178148943327599720938747864878383569629434032644490438253647/24283654761251669077105409814988228211944623880727)
(1618886473395097711384082459669776367056052578176625731419681/51021507204900160597759648998400, -1547028494992202532289484495857282150989815142031670923926037096706427411165263776642155361/364443262389968328200876494869078547874148352000)
(1971345607257462853796488994133330469063480289845207352536132129/70940950648580001619402636296958009, -59896313633617902582346202754395658252045908284557111419837767016348325856375456098858805690785/18894939511586109374606673628769400271318916556811027)
(4491695403583397288660808303444702149556316821540598228643743561/1752384105097537051604288658498921, -301019364241743075886304929214947307983828593220329109297512924809527233968111088755282425624693/73357404669942027145504843954961401728225291222219)
(312288049174592873059055855376555265662299555200910039758795513461/11365949799564982062940494377678214361, 118303191532359367675280090801946477667429532263557520219088981007785893116403626396193957757701147/38318508848966989290150927196884351155590742423285605859)
(177539678677887394666473108231451831940764365576026731270560947727301/8031336556199242625723584649252900052441, -1250651155591581087488383612982511826854330078765796017998297513802121020026258925841551694105661501957/719750107381877800226717559420295789403968064231628837991011)
(592078613400141085975432264304142822547372147940131337631897316928543/152094651703495535871158929177964640776281, -117125160170017258589641502483510000282978833651634665591521611945627911506981665160807686831500326021029/59315871274049160732091504093358870715087201196860910489585571)
(-1231461170780874525625873080735225790276827881906729004667230965411659/45936015475795308012363954428578661818225, 20095946281788929605147268973299327408424165702621225046164733563195542547043822842534501439808321905409/9845323303633219873751540016723520511271969751636878348991625)

Commentary

Specialization at T = 236253/205754 of a rank-17 fibration of X1008 (2-neighbour of the one in curve#751).

last edited by Matej Jurasić at 2026-09-28 18:40:09 UTC · history