curve #4957

y² + y = x³ − 24440149608834728127954520493046376

a-invariants: [0, 0, 1, 0, -24440149608834728127954520493046376]
c-invariants: [0, 21116289262033205102552705705992068648]
rank (lower bound): ≥ 14
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 20010625156114155541673742142742291061353130759380219529827363403
discriminant (Δ): -258042634373760903787590115252203953819936861371951316678811732570801243
Faltings height: 12.3815
naive height: 171.8862
primes of bad reduction: 3, 7, 19, 17431, 11742876743532235184470771
regulator: 3503616758959800797841436389.00658220603157086462084887037178581345
submitted by: The Wren
submitted at: 2026-10-20 23:49:02 UTC
last updated: 2026-10-20 23:49:02 UTC

Witness: 14 independent points log in to add more points →

(46336652099987234392347218835260912896908267668717159922507298767352164769624/81364205656587046031419398105492390699233333118806928880623473025, -9291084771819335568279344784072568017130718086607093481378460370560010380040753235646938112390867271295619742196568/23208660908563628657876706246148488318666853348196324550779062548658480151323492599053606578941375)
(856345560199552672677655784441550730949304445786507738053208592521710770857/211680863042841607835159124521092553477089985249533273481973924, 25054941969329276775989337412401678737217140267395251458371394882718302537332248739601592158811566571142035597239/3079799160476149277188171640216278025660394490415301585369298177953946667184316503288008563032)
(13073239507980438612702137446374079249119074335174545950434/17419257426748317387161719786870287561903146025, 1450917904787596126074032844976089666622336549797398138030830064536567734356896376289248/2299029034396796067975918337301915658217146299307319351698267024295125)
(880994002602608900416290131585843915741349938075184283713252938485344778734364/2618102161638124114732195802440837359337945148778269920086379113225, 495165553388167523480470378506077497203884539106340528174667275303974848181769720915202817989437488893948455037279112/4236233437652311577943730528571597024147781184800198188195813445371398018069447580645132685349822875)
(1131761466951789098778734130019185382149922833268561423275063858852641302/1515369872709371890167160503697242930485630948723220509536089, 1168164379331816071432500153516417294965456728121264110494134725994577039554206583717772551322897175468250704/1865425774481837189945916385008186828877609440083660020700619307671280267899996396558737187)
(22225660527956548980213124077209147762368345147945235407677196999481807671055914/21337971661882651087322532704018712185598592207956396834786609005625, -103641624585643697502781012596386998049003777671594579822755567479774967327739223197513407239744725525079006499685962413/98566582971197862495535135499679017765761837322388187991386051466002310419609401906675082381423078125)
(289917633431907394481790234593593740034354413754560655733954873630177986300533463201/250565442650103217517767826375525263864978172403315999516197134316384400, -154866879794053465565672882922579625116257851430303111284571995720834208672250125372206906826336045737977692032490314606870801/125424321691303725535277657284215095419750716513015452279814826972504694315044368076452034538591785881672000)
(8258389690999430498294716987262350415034132763239798795918349936920563919952/27302841107983985811105463944815102739235288653536495877688346889, 256526204739584925350890295496979611599019683143711867154270059291315875739173601251954081153131688218921258515171/4511404438666439636137820160802774452538915169182852857297783127858228397670638750415565351098213)
(1357631704103629189755883654308838015693919474838608033537197371612878521189791337/777521464731645282105530078810298180771072152201283458572267963371684, 49908413724786084847261580768255662111432765670218645983777489208160548490589026784902546948553895223962349852702936635799/21680464412529912996835652239633816536077958682249495871155034557414422019283391778535440453141761169752)
(4286740264015597043121090584713541287167751700481837922841594444464/1605140855542555633597361485513605793111549866612515625, -8869761738764167934652603764272721326040326878493158817709513774959076133664527176606509046958524813/2033619620954419555772571617944362813192085312609521629962493438417962958595703125)
(13029983788788437132978960659533725826204409827377712797073105128542934023001136239859231169/10727905651208948451537215206789442528745698088103969371285304638384696630260100, -46712544048306044096545619652223025170035840783619805241667097057966543669226226417656114156639424299344071847581777135248348697100181197/35137626169619564464811611780366786644156253488319810286189412001099446574379347120222537815060587578188385827917349000)
(1229917338674986558092542145863578541371817826096026662038872790267465022/336398615382471452391060641803398609485215837028754013218129, -1363657382631112887984103754906339363308309768693330757077657064313103927377273708867704258009743664441968361/195110795383989711219576871696326747374488077386648809502806773798610437327571347488470617)
(109075395778875116419881485780764440725809064719958177335295649815850700039672/371978959582121299621862806135758314393936633740980721580197773129, 6307211439384015786950062710327338680043082703829396627284285760953785832840116625193502612698787932635004265736376/226870257024758089196547461008232858118804597909740990806178771745907408241149241215624910711288133)
(108132754090067056187567720460102695361479025699065708560849/40569505284441061259382501474104854266030849600, 35534928000487890829144579109446287058344514679521184300425666959004600744512426591519257/8171458280430548971605372759521672104190379355609563303747698324544000)

Commentary

Obtained by a rational 3-isogeny from curve #639, submitted by Maksym Voznyy: https://elliptic-rank.icarm.cloud/curve/639. The source commentary credits “Reynya - Voznyy (2026)”. The source model is y^2 + y = x^3 + 905190726253138078813130388631347. Completed the square with Y = y + 1/2 and B = 905190726253138078813130388631347 + 1/4. Applied the degree-3 isogeny X = (x^3 + 4B)/x^2, V = Y*(x^3 - 8B)/x^3, then set y' = V - 1/2. The resulting model is y'^2 + y' = X^3 - 24440149608834728127954520493046376. Fourteen supplied rational points transported through this map and checked exactly on the target curve. Rational torsion trivial. Same rank and conductor as the source; conductor 20010625156114155541673742142742291061353130759380219529827363403. Source-attributed existing curve.

last edited by The Wren at 2026-10-20 23:49:02 UTC · history