curve #4022

y² + xy = x³ − x² − 2287191864007369282657909564252676904x + 1330362622907855018752768847663776888265944490717557460

a-invariants: [1, -1, 0, -2287191864007369282657909564252676904, 1330362622907855018752768847663776888265944490717557460]
c-invariants: [109785209472353725567579659084128491401, -1149433306192386735708358841755911466407667574101391434149]
rank (lower bound): ≥ 17
torsion subgroup: ℤ/2ℤ
conductor (N): 279371967069651688648184842896025457250762762386873734247421810 ☆ record for ℤ/2ℤ torsion, rank ≥ 17
discriminant (Δ): 1169787669604983392589943132061668014738167621150030551850284500237937965631973637474161448965305082826937500
Faltings height: 19.6616
naive height: 262.7748
primes of bad reduction: 2, 3, 5, 7, 11, 23, 31, 37, 41, 47, 67, 97, 157, 239, 241, 281, 4093, 4513, 8093, 45853, 314903, 743923, 190467271
regulator: 13904094091008699312191589306817.8592397894555608686222346713019328469135
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:47:37 UTC
last updated: 2026-10-05 04:47:37 UTC

Witness: 17 independent points log in to add more points →

(3252589155179775727235116997612202843881/3608060075682529505536, 7210869581808408974752524050680417225426690779596329414243/216725812690168959163139078262784)
(155292711808277721914510141280690323967701/12432973569484812026281, 60767301207718085686906843026825080027618997402747439668959433/1386316893757145531951035880883179)
(2994714814275811620028788298288839586841/192205092848979978496, 163137850814802606456323937523072280848358414333429749974587/2664693953405045498234014355456)
(5982631303337752173931257053380603499/1269213669060007225, -13957326505804593876183891755344499245955178513148045763/1429888294860734384740457875)
(9700728200518729009688459225906137675057677842994662805649302103835615/8984476457131941747462628565797935522760358875121956, -294475700357931357207467283977140997007993550720952273679744616268314053859962090213101444383286839240485/851606882201252704256979480267198510247815838700637989470908474612955317689896)
(387498032291248859121957572868891788120584368441121421501998814509859/270396367284959504331304828546623678799956978465529, -4436875011570190751370069759933271363453717465315086479092132406856396439111924261067888352247921158259/4446325769012280006237882823381168376985405702144505644658658016510742164083)
(5682696928190513628680574730097008749260684738314030136885/2804930620809525718201554411859794129529, -332582777162191138476344749697373103107239467212362956027654371158797153542964954062785/148553601551391675737207446836006786023196421524663064912333)
(8956389935593785612093405585470288680038773760065/8684745044998581140220917334016, -6695219361956852759910382213544173446336543243881090818313840029233534625/25593849410436703909299248392401894743459495936)
(21968284877177436158997025798908057536117289815221159848746974799/214175383926015140160510555135734219048319844, -3255721769816933058298727200534218883531519001238434877494718221917832257990367049410832464104931/3134399368273752635425176168860780811522711302942063407814992123672)
(1060936785003743274822569941628096215109693634158639/507136896080096695164803720054884, -27269244839900747411157394219011265497218512060780395510311027557879235668821/11420570848622244882250862526697253055782712114152)
(12341379841711448191069005921208905368485940082599125413374951/6362008471235792432344691553647990846669956, 32860123725815270538354304161150228894354192879644834009037931668724298825801071593646849317/16046908068232146079898783311925072648102057506943635436993291896)
(3940294050998693338480131702215116550925033865692943039/4118656024948812687043373101936952484, -1116645819110825797664485736204429943278054128482505525676497732557179528538898021/8358595026242740165768045228580871786822527308268573352)
(21592471593259563565130887453905298038131315833091663006374755228749/4492460135361139385347808018750564242345008285169, -95870030242861048355609219291539767828240253073157985435026473425129656264652109161951221005533963731/9521959896134548745688496146616571908172520841201681763469856858951403447)
(72625955215961785024608873913530292238398020377824042469/73297815894349736107582790262957247225, -120526068984956231350238585556689257964461011622194445424136882450508749963061689633/627532973771706006288508103991652275573356808462026442125)
(78714924678889950025321492635764812510230411541861651946741/83666010982276120272387676575297569720521, -2571121542650120271049751987783293677696168626764206165627821874416568687170469871979513/24200458538633315951503645293910226404027207119311756546754181)
(186371517047894021416830562383088101052684499585288225258985859/16536346302593309293888796316040498745293329, -2522487794431000674598504434205215169842429694385823586810938244531790254530571204980900133591/67244897819962946609423177612525993306949804149497788925337893767)
(22074426249640752613888400312133947121242675916964160869362374053636418589404170160286702393094049/24724872322351381464296725507185883859764190961727383238304788477750364166156544, 154295224876018538375642547463484833271406554065835159207579786775736612564616632925163730295795861599207767687963228542018555199934950092421431/122942230016699904458540558977420840062307938371806172061254115250065019974547945206412928297662310912723618481161596928)

Commentary

Unlisted rational-isogeny neighbor of curve#3565 (source contributor: Matej Jurasić). Source equation and witness provenance are credited to that entry. Exact rank 17: 17 independent points certified by the unchanged ICARM verifier, with a matching unconditional upper bound computed using PARI/GP 2.17.2 ellrank. This is an additional representative of a known isogeny class, not a claim of a new world rank record.

last edited by Odbayar Shinebayar at 2026-10-05 04:47:38 UTC · history