curve #4959

y² + y = x³ + 3233989876366401231297754937582669

a-invariants: [0, 0, 1, 0, 3233989876366401231297754937582669]
c-invariants: [0, -2794167253180570663841260266071426232]
rank (lower bound): ≥ 14
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 116790896991187197972944027924327549003781191857801594054651281
discriminant (Δ): -4518154304830240324391230912068151441335085594563790792747972876514883
Faltings height: 12.0444
naive height: 167.8412
primes of bad reduction: 3, 7, 19, 23, 53, 307289, 711959, 101560749485011
regulator: 805348304786457847735130713.348977402298928468503518184462239474130
submitted by: The Wren
submitted at: 2026-10-20 23:52:38 UTC
last updated: 2026-10-20 23:52:38 UTC

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

(23298519854634477310783757516001/281586804604742560000, 291295903693061963453828130253228818649554007249/4725181003471225397619904000000)
(2084149720123494180166438748447809/2292456943628998322500, 95351149234966665840710721901613414987345828442373/109761941766422939198606154125000)
(-3902835207133065728637169689819647/53687900359963616791716, 664085730505080492431161567409855911309240510941173/12439835278272244989729673173060264)
(655955157108231087364415307232777/10881291107144564816644, 66699509927944496994996036229473240820263987160401/1135064768421772010519638260948728)
(239070211220735300215719700450690753/488194750182158041232016, 118491518948308629388916041512899680118922819593503937/341106234618456246857134301456607936)
(47029370930757333031128389241/677175436904890000, 33290669871665183451049224845841624147840211/557252204104461243763000000)
(-2246681477933983102785626483999/104995881463800040000, 61089824952043283999504010881194459991586007249/1075866527390016767109992000000)
(-58250762867036619149944745715951/5615963911846119040000, 23929347087310659925271556958322821628511263513593/420858848446504304005278208000000)
(73320265421575413848152176313/829093916134851136, 47299720678565838745942108421893422523677523/754928102531664858928592384)
(1189167467306827242249917454611943885235332204356437911212784552181190930247111944045397297/3088440044860136615398239840277576272404054284162248727978874279319457910775184, 1333002111250820234119649935338817904608693321186870978360416583077023166567493249400392504916437723711538602422802004787846964959781001/5427611604690011217449773752409252390056828797706208762070142066591549012815356172120781164217221163498065274313533248)
(5537074317197968475626822186955443473207603987230685548486755845855858064639168718442611972608527777/446702487698383044285540690060874434987945618861859376017455718698048533649566288387344, 412022466430180769508566877398807265916731063665725192171529766289401885313656423691573351064075575346515823874889847902170278876558344547876221061359/9441207802179847627985198743568048652466053331360455666906716049779390239737678527180999696752424751438373154263035264679499908672)
(80563515314176546350512675412583982440916850429559354813489539585918403239026936678175321/1617351876590914794498423971709337058210108029978150325312516908173187900033600, 119184640096661605027091334573576135493859067931812305155485266962799319126432998884581646385099464285857810965673106665491053798003619/2056869674343845881873533212784135301480044331750333364347207280770777809952397127086007025003993725477992070180416000)
(888424150982741252737304029095750625142516524957691393073459807005963704075632416035271291120023158891240961/8894444701794238984430316123099788947747045386410528294881520549698425726673190320525946820264900, 1725348371202966535672476899216867844660286979636489800823644162682458907728549231838342202718087590248085598455983995702702119133161746588760579160390754928725459/26526396632668557948728102654744462988629233233874589908500809743869603313473978155067260482698557407160388352323033336515762911520005433367357000)
(-4676659680335102604064026494407699326349564390054407256589971070020686945328872214692939069071010543756214263/46168329185556372010071233030926123619925641907080443206657982624612603677375338150933380904489764, 14695852671193256678729868366254131093567999792400526861584452780905292308633984660871192351863156148453037490006347658500279685753382904132997318737406361858721441/313701241040103775850992332849659541453399425463688976498651010137432223239242684473029109934665540133182291719506656244896688793667261264363568088)

Commentary

Obtained by a rational 3-isogeny from curve #4886, submitted by Maksym Voznyy: https://elliptic-rank.icarm.cloud/curve/4886. The source commentary credits “Reynya - Voznyy (2026)”. The source model is y^2 + y = x^3 - 119777402828385230788805738428988. Completed the square with Y = y + 1/2 and B = -119777402828385230788805738428988 + 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 + 3233989876366401231297754937582669. 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 116790896991187197972944027924327549003781191857801594054651281. Source-attributed existing curve.

last edited by The Wren at 2026-10-20 23:52:38 UTC · history