curve #3897

y² = x³ + x² − 45875433589310492345560963870813242004955686812860x + 119260020574067829647229744987894968769054120014581280328759103765251260400

a-invariants

[0, 1, 0, -45875433589310492345560963870813242004955686812860, 119260020574067829647229744987894968769054120014581280328759103765251260400]

rank (lower bound)

≥ 30

torsion subgroup

trivial

conductor (N)

35811734834840131737951289120466348118968103834120652295189080842675671967619553491806095234338917220648409352052937056440

★ record for rank ≥ 30

discriminant (Δ)

34717592622529766056659137031235917638613704361773491990925016995264833275804710607257386735288504421736919300526051824158635118498472848624084000000

★ record for rank ≥ 30

Faltings height

27.3680 ★ record for rank ≥ 30

naive height

354.6636 ★ record for rank ≥ 30

primes of bad reduction

2, 3, 5, 11, 13, 29, 41, 43, 61, 103, 277830136730825736869367244625331623089, 23383598959940946938756969654336040344691745139782372447555562422318291

regulator

3242916811710520510543641980274287.1643079346121661553245803692051846285625689592060961839011693959

submitted by

Ritik Jain

submitted at

2026-10-04 14:18:31 UTC

later contributions

primes of bad reduction recorded · Daniel Błażewicz · 2026-10-04 14:43:50 UTC

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

(24053341257644574542875386999034775976582415041525653755817620/5393524061304566061855080029739523289, -22986238170432946511050369799640599824401499769606056160819198882167851542988902058485866500/12525899728299166119836623407890303539640153965259216413)
(-45857984959154776095767066/25, 1755519123308002152810849695932938817398/125)
(-56210549853887805711802327820/12321, 20902718878768316729052100627953561801912500/1367631)
(668549596265227127791697565/4, -17272257502062359522168626304626776077235/8)
(-136237188045053131679965595/36, 3336986334556195131939422198676755151025/216)
(106817303737907812392359175664993751380/29708833237561, 144504188612155903154292325950933460354837571351091937500/161930401976818472941)
(3671448209011183136199430, 565814038750983861008032947914608350)
(28670484760528173986200116820/529, 4818364016053668163976319093023928243601500/12167)
(15601692339639549908191559470/529, -1901319176276316669028644120588077445844950/12167)
(3739318447475486691695230, 46848979451145009999734418961499850)
(85186765962986625179780317/4, 750524495052655968663530274598001501907/8)
(40379487001718954580947718730/361, -8099572465669608200004585250390580062773150/6859)
(-99425465587321878060492995/196, 32744652846616834405338662822241811200675/2744)
(4084669162306754679026278, -157495592319881980278622743997238334)
(201620041615699113081985570/49, -137769117492078784692785201182223273550/343)
(1783926233851486476971826295/289, -41479836848530949636912238131691135898650/4913)
(1475165429548226865271158416665/399424, -114837130175823638257469897917199589536073885/252435968)
(438430907020928523291760998253271215235047133202314143676093535581721950084114041720/127329961553125486839818282195501814267244842264858484119889, 66191697427636452137633841091958490378123619876209524910498299737173910575322026894295125051789465610301608286869357852182300/45435562798629840630479701755012434610995150361154220049028777456621534925952892686234713)
(672171988614133241414851772295481562453760396668544260078606432225647310992174828614634137361045945/544791914441012410196761908062352503705511057461509465419178921211379180544, 102152419734855827365627110865344099824893506532507526129123047094439558590249182743654539776098307882231000728409694861774814503798410158456086392275/12715867093767569011953788352075627946552805064544325519944666326621938024390875313314368833617593702655733628928)
(110393149403258162037405489583372952830/405521282478481, 84389674326958417981522611703448943816524601934331020170150/8166208751666077367879)
(9691773659255844692015480990073995362524701041989/2654250709034643135127876, 2847029158212993964229353143092193549018501980959013351761060956287676945/4324271098599555794106786151101581624)
(2000409450927033734374037616736574930853147579670/832504120848489224889009, 3634989738124969502534653258311801412203068726255855114850027625794500350/759590610860095728102878676384156777)
(571194560315290700336528912864605319260953304230610/165083067772342415527294369, 2963670232973969449369064230775846045573872286434505978136592221849592965330/2121064114190182262255093789013456312847)
(49342420650710932315538254095684288956936397240584398007394259819157295316/12077152210380883094265604655465363760452757679369, 7111199033942895858384997453202230949186978070437833947212697168612245528349111708801642346357105884275145060/41970757710799819363499397099621645354649294887655616696064257373753226597)
(151247141542245322109588624773773803171031621341728779512887179335380/35797833801274865406737239971245012047451281, 198102743401549168095261059216166608140149568401490013260431514260175861898426888402444209122472211100/214183061055614684325206759205389250924048482849023567200628848071)
(2659484028345823420865390732842313192766850041433539730/292277980581147648883416959881, 3371262722973644057474370774746805540877317034799313470860369628787851989575638250/158013483353856961922733152032319574964386779)
(-2761439877647308988013583087626121106816290970448448356330402552741127456227981702947284881108297595400/3719830091739040270629882073337124737712879145442809918843794726352784462640409, 88715214663585017711900604035607289078741196567001005441557876138322935132768221991264058248446612149054505021807747676639143513666121543943205760702584660/7174384611012689421486387857533110988339959486120984336077855448770690284019881205411804904868466072729425608983441923)
(-718010279488205086182454573334652508571990916656120938809766778/194906934394116249368715929601126617929, 42002166247143251566359675100739130771389099251597545131712271165569665382590341648573668397542/2721077655453597082778902257518005165571171786931647236283)
(20117259375954089969209918882639452967584225959264613467216279287/4740173739471323482889147978856766865569, 327264104959428503995138486593168866174204486297603009495133813064998567269506345604296474239462/326355841001933670380279876563677513356147062433721103408753)
(-320773994713892784682720587299611788545/61830999366361, 7172436292909895010401488261686166637408392067688318196050/486193780556526094109)

Commentary

Specialization at T = 208/29933 of the rank-17 fibration of X1152 given by: y^2 + (Lx+B)y = x(x+D)(x+E), where B = pE + qD + pq(L-p-q), L = 266209T^2 + 82936T - 341948, p = 207259T^2 + 5429T - 17528, q = 23(1200T^2 + 4964T + 1995), D = -6676507890T^4 + 18743234661T^3 + 86315454132T^2 + 26986412517T + 998361000, E = -345(1323600T^4 + 1949872T^3 - 87498021T^2 - 249387955T + 11203920).

last edited by wgxli at 2026-10-04 17:25:48 UTC · history