curve #613

y² = x³ + x² − 1107604775133466522261720798327392460023905554580x + 432991693500195656777860504330355438985085709768269367947970962468463328

a-invariants: [0, 1, 0, -1107604775133466522261720798327392460023905554580, 432991693500195656777860504330355438985085709768269367947970962468463328]
rank (lower bound): ≥ 28
torsion subgroup: trivial
conductor (N): 342523533647914951820703775690355483082106333747339519016573360293993471250429399722837611188967052860640860844920
discriminant (Δ): 5970843284047983657820830209750567428121315734152849308437287912124640215147333552326773197941175373732840335316361799460887453328243239140000000
Faltings height: 26.5466
naive height: 343.4925
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 193, 317, 1303, 887859288553727082005448750047170960641747014774727184925252721465916328351221746776166724561
regulator: 5702925797633015029760341179938.213470004520533019286848244792643463013487886474216087447756970
submitted by: wgxli
submitted at: 2026-09-06 16:24:17 UTC
last updated: 2026-09-06 16:24:17 UTC

Witness: 28 independent points log in to add more points →

(191232322533099048737012664398996662039186/42776319612248161, 81484028748624098471481571914145291374914423835198501945114750/8847185311945550570235409)
(1137516270013902208461837847278/89401, -1209181937956367801065352998213154673180028218/26730899)
(167580596206580306017131762755536090691948867626066/338580388948746611866994041, -483907330531357540152459349787306424953278760067824188471920415886251838250/6230066743457656322598608175435673532339)
(1709269707113090455274340051406/19881, 2234516166073426916736421792056689660142829750/2803221)
(-15293293150933214489788302337759328516/82085466370609, 591644352094562397264839805886903134191183746345328073600/743702780120753712727)
(13776208082435595676071249987729785926/18916401791401, -9239946671879800798952681477579200702587311524372422750/82273087394938577899)
(4427584826702559086794985356/81, 294557713830455257684865787165083577599000/729)
(2057726845448849370425875486/961, 83620269394900506732701760043039386257250/29791)
(83037960142816719579296434/169, -188924172991322197495652128364058724410/2197)
(499088418427240919086311082770055144654/607949255017129, -4158977490239920437613593653421338836365817179976011333450/14989978015885228363883)
(10079510109718769438276026, -31832563036277999603348406473390475750)
(2310866820989371891811928894254872418014/241558955055889, -110440143716266273522055145199330332191800850541384585649650/3754349619824121171463)
(-1143631814615909754755349, -451591491915183197365516137282604500)
(861660338452313404665631221096/373321, -727505905770194902555808662424502761239692000/228099131)
(6178267917360089027545276, 15146634711511882585220779227143628000)
(116583449580810165409002294119978808439/31242141344089, -1213149917830075893626920071169267096676615970047289147750/174626918052121110563)
(37512622732734354772369763560065426923509/17492558940075844, -6510886224771129136645067641186681414210138075447599339347385/2313556014596306804815672)
(1386723292323305265923692445140416497016720058/1897824667758047253289, -9711480769790121895810090641299982560463240066607081736206123691250/82676889844302288756562771331413)
(45363681187298092190932825951677828124/63034278784569, 46451787323526955176223479797677804123724460346991335280/500455172427831317997)
(2905539420009351238688489122533024334/3700879470289, 1548883316100665777872850012861177789345425074915527070/7119629795919458663)
(622554644491634854469707725007931381628691/360706268860416361, 414597407334141225878460629114633190921703666739786097781421760/216635953632326064872269109)
(480146175369656350607317588829094309826/535579981045201, 4966383107030385068860573378671940450085013445038691717950/12394713804916649707801)
(7230179903231891022660854721691390219402452526/8752272049968555093001, 234121839632198699389790509538417806817681715139156385093481293397750/818806370271336869374675298985499)
(5754516369680649097789734033741873106/7019802559081, 5124017608381952840854457291666450855616516103826787750/18598903702062077771)
(3021428371551772725706694039494/1026169, 4952433585625941469299875196787332938952507150/1039509197)
(5639026122978627029473186525500010356782569681/49449069004810240000, 423435950481719190907669092305965741510173510123069621356476441920921/347726011478846423072768000000)
(198930838286746374740220696081486645021089074/121207752753572712049, 2325161676857475570137504613979639157708071750232197674651954040050/1334429391417933263036217507593)
(2713568571822601099620819472705026098715915736/1621575315901933912881, 118001469140104947448244377878652145063043673024410108016535750854840/65298873204233543936609732952279)

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