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curve #273
y2 + xy = x3 − 201769035260418549083594900060734240952308696994802735114305555x + 1151107939141058565733479426024323225135665982951300586808823640527729578307228357301072889377
a-invariants`[1, 0, 0, -201769035260418549083594900060734240952308696994802735114305555, 1151107939141058565733479426024323225135665982951300586808823640527729578307228357301072889377]`rank (lower bound)≥ 30conductor (N)`2381958488309327728488641214562148525681586925734398576894288277390640894675828305511266759053188773997283310012808983216600083938367948090232306090` ★ record for rank ≥ 30naive height442.0854 ★ record for rank ≥ 30Faltings height34.7705 ★ record for rank ≥ 30discriminant (Δ)`-46714661255308767314567688733841531918983356002159772613256840842851650254036518701100578342601553513579222272710220496887616034526983492843954090554197033638137245037791044053017600000000` ★ record for rank ≥ 30primes of bad reduction`2, 3, 5, 7, 13, 31, 41, 47, 53, 67, 379, 4349, 25721454817, 97018222656318846556561979214040553412450110580812087282349817173780902099339117104673990259247421230916714670243202937`regulator`10720686604115654188358042985326663938.588084900092986474567459949780498792191278825612795832355129`submitted byranksunbounded last updated2026-08-20 11:24:48
Witness: 30 independent points
- `(-4761204159891138283979053265906, 44764265461782973805868732003346421827415264953)`
- `(-14158422539541566469588779426546, 34199834254251713784176619895082644508395077433)`
- `(-11522667358396562420423130332066, -44115070023357103726405378140637465204943359607)`
- `(-204839531927226269712122049566, -34531574232452693997231136031282772551453427107)`
- `(3899324051227528532535432912094, 20582352852872417675268569815574934013539218953)`
- `(149851368287976334870008075289384, -1826442728148288630645637436047625928557963231657)`
- `(240440240734591134232325971191694, 3721941824016160691689265341458606456425791434553)`
- `(58446054919170749975942104376446/9, -289145377197241504032247540119122580900747897469/27)`
- `(25642661602146479458845459929344, -113306861325798987289137854129016658652160209297)`
- `(25720885078613923889202869994094, -113918565504468051036791617945007239588074855047)`
- `(4956414590296956229584100339596814, 348939117745197814060339374812186839746231405619513)`
- `(725964821994104294477684670330094, -19556488133953913131900670560396205869420775943047)`
- `(20802191136944676997135829374, 33866070189878993817062821320678356972094522793)`
- `(79052318332408565020526148386446/9, -202982221452031541387733280916787231176177841869/27)`
- `(-11232245340662775388535509780886, -44725045659073489550941507272219743508825024527)`
- `(8362456338772815315335239525614, -6972475614865802969141741730862843401376795527)`
- `(2011658715643038193607509024534, 27447371869432010931671648582500375378543228993)`
- `(5027695440284894460797358334207726/529, -116678851641395817353208818767411586490893208148849/12167)`
- `(24649144267565165528439068441554, 105612540318783792474731275264940719325335867213)`
- `(-12211389420609043025008816968566, 42356225616159991618318584560811156010370207173)`
- `(7798692390172953821075781106768126/1369, -691870045568822811690292896396241871567834072004011/50653)`
- `(87157992815740534253438806045216/9, 277139378791840529410298740802253693668472375061/27)`
- `(30786757706172245427369935940751/4, 58841476683002984849182029306774218124047405249/8)`
- `(245309280348041323814668746104926/25, 1346501028820415725958868015485008289981037919061/125)`
- `(-343878076324392159036619356326, -34934957027779219869199839566344035316624147307)`
- `(544211807917340289404451270094, -32271721754226832038590040491036826507284103047)`
- `(20286216384652039303944170492166526/9409, -24593234902246769413006506020777691495865223432164871/912673)`
- `(-25558163204018019740775243468600589874/1760929, 74707049582033426178338768659390679551818201954095350999/2336752783)`
- `(4546264873863829383537112534021848799606/398521369, -145386763829319577901520209264368135012688669041886277075149/7955682089347)`
- `(1709164065046406773620054102684586/169, 26450264171408287955631955124255640794301463854841/2197)`
Commentary
Note: the original submission silently dropped one of the witness points, possibly due to the parser bug fixed in https://github.com/icarm/elliptic-rank/commit/7717d4da140cd7ba4ef702c6f7841e00f3194c38. Under GRH+BSD the rank is exactly 30 (if one applies https://arxiv.org/abs/1112.1503 with Delta=4.25 one gets an upper bound of 31 on the analytic rank, and the root number is 1, so the rank is even).
last edited by Andrew Sutherland at 2026-08-20 12:46:37 · history
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