curve #845

y² + xy = x³ − 68906458325887563494253912743860596601427440743924636890868535x + 219726256542658179767501421714589587209898779390156657938983977041099527263435646417385213097

a-invariants: [1, 0, 0, -68906458325887563494253912743860596601427440743924636890868535, 219726256542658179767501421714589587209898779390156657938983977041099527263435646417385213097]
rank (lower bound): ≥ 31
torsion subgroup: trivial
conductor (N): 18115274159851185649085973860282364474114939199999740764566104096487265415127779932457830911546742604761434655236135910258175039596245279017931519045110
discriminant (Δ): 82385085647565163089469319506325224577187238515665763733099256336254535483409981177829638967597372480839499895297313893579274108685651763130270268820454766217537536535157995274240000000
Faltings height: 34.3636
naive height: 438.7772
primes of bad reduction: 2, 3, 5, 7, 11, 17, 19, 29, 89, 97, 179, 528420985369, 21835888829059, 609783941000453071688454628296149076008452259419919624936352567500725084769863874156805338122461245907299662147
regulator: 290374633337034763471895132770238785775.0224607139527573559911137907928727556081032817214553973615656
submitted by: wgxli
submitted at: 2026-09-22 15:14:13 UTC
last updated: 2026-09-22 15:14:13 UTC

Witness: 31 independent points log in to add more points →

(4492189723065914919256867882934, 914724714768630901238259806376949142963232533)
(4567121567039911266825199317014, 534643646961232792435902595697216090643639413)
(4586421283438238075388908357654, 410821791963280157575133208491598963338280053)
(5037509468610356621679378871334, 666098792397671427124894554496634456754265733)
(2659630394938362489510492764438, 7434634969447538378141678087206140041567980661)
(3663881909579733757226379828044, 4055267373636724119245464691819620518768714863)
(3677812647428541656906378144054, 4006049353458969139310786882901444065827897493)
(4098341294835689182470629375254, 2482228482598568995384324738277746299302465653)
(4164965728556382418786791819254, 2232180041009927043930804781152182748288551093)
(4210076388396811999333015844534, 2060904208652281762820569092863917211521306133)
(4417431261416015121795299132054, 1239813367034742170365682325542949695429486453)
(4431929075218277083918138761814, 1178834426993205829246391686760443853789620213)
(5105676126910662784165003206038, 1003293868066583439325644633458295443785691381)
(5176341638430813588645978697934, 1319257072366563187890692056699670017335927533)
(5316559979035457788289474404022, 1912540767664534490942467385279679856808556053)
(5324224606886431739743202668022, 1944363137574878874350541396985641121979088053)
(5482287408296083204197995043734, 2594995395099938359808519242326412410251570933)
(5548697434495430663915305345814, 2866828288411478685823776455429234788876484213)
(5615878802954628697457810529302, 3141664450647791215948876141974015871913938805)
(5733477463417257660942569682134, 3623241806674604699060111526578454755278054133)
(5874686722234872651737487188438, 4203481938879821615647418800850411741692040501)
(7017936048521021994702580091414, 9043699426822951025499785852469862997523101813)
(757752158220185258826363604934, 12959449505271470233127111406483601080657066533)
(973557344004335969787009680534, 12392119236646578470308138231491128734565177333)
(1013068729907318835646725610934, 12286537312070329678298138251534690134629856533)
(1639444315586958180395107629614, 10543453748450346001731175965059489249608951373)
(226164351008217133335210359392886/49, 36401096114268784547188426613387846966845341059/343)
(123713595815943158865232342397366/49, 2697460873874820084336950933813275583929181950019/343)
(3708780681953243884687375900371494/841, 30996726003060644535638723469585545194108289295337/24389)
(-1732275322545229786628751579745114/1369, 884459401907288495674534376294947971656404061298689/50653)
(3228375666134932050157146796346896575014/650199001, 159093427395242901986547740315901913477350720665857393087/16579424326499)

Commentary

[edit by 7fff-zip] Specialization at T = 504307/1742937 of the rank-16 fibration of X1092 given by: y^2 - (Lx+B)y = x(x+D)(x+E), where B = pq(L+p+q) - pE - qD, L = 102579386T^2 - 110801088T - 18001125, p = -(96T + 47)(1094441T - 760632), q = 101(163604T^2 + 454821T - 85571), D = -(96T + 47)(7477016504034T^3 + 7131320229659T^2 - 30633770734995T + 6299022506152), E = -101(58579461360T^4 + 18954357876307T^3 - 8762256702014T^2 - 15609000269917T + 2960508883604).

last edited by 7fff-zip at 2026-09-24 23:36:44 UTC · history