curve #4274

y² + xy = x³ − 1455949875263072888981648277783781400x + 676887690880153858532083058848946346979072317432975936

a-invariants: [1, 0, 0, -1455949875263072888981648277783781400, 676887690880153858532083058848946346979072317432975936]
c-invariants: [69885594012627498671119117333621507201, -584830964920452933876548153864430891796597158262523469505]
rank (lower bound): ≥ 24
torsion subgroup: trivial
conductor (N): 5500829082348913334217047404382743088488241899479066241289896407758721932164790308655994186
discriminant (Δ): -408725930731533853382497009861051123524424125397615167055338039840094902916176546130077216123842160699998208
Faltings height: 19.5598
naive height: 261.4218
primes of bad reduction: 2, 3, 17, 19, 23, 59, 83, 251269642339847, 942198604500147520764311, 106447130856764071772177688995129232648180811
regulator: 12514726908117829930387793.052267157394625951682399992759399364657924414320962207963398
submitted by: apkorte
submitted at: 2026-10-16 04:26:21 UTC
last updated: 2026-10-16 04:26:21 UTC

Witness: 24 independent points log in to add more points →

(695635339761463408, -26483515431820779728459096)
(764434110277975066, -103025651002432241060107772)
(-1180925369321652560, 865653135618883438334872936)
(-712249781820870032, 1162997650572810649008510184)
(728099626688072548, -52895232235642725757438172)
(666502832526304144, -50703840973414317574551320)
(-656117537491155440, -1161770094276406898062073624)
(-2182102455238917501/4, 9152217489413579790111443703/8)
(751289945011676008, -84277621446844084554942296)
(717054993205049200, 39726523239745884361554856)
(698596276414543648, 26593060068359888297319928)
(635346882079646200, -91226071219683572740106144)
(707781622800015520, -30979749301103414835098504)
(658936198169788336, 60147141149661785736842728)
(12414194722011054064/25, 34540137838369775896240921928/125)
(-511415056363904346901904/395641, -157208258707076968876081129248983608/248858189)
(13576932321644246086/25, -26915887529463567672983442556/125)
(271791311963495140, -548862941980686041451619976)
(709715720191429762, 32533556992829676774908164)
(31703839565242043920, -178384716599488609484264468216)
(83695593328297596592/289, -2597522444197128307934866031896/4913)
(29479361345436948880/49, -46921592741224986778008867272/343)
(-1165515581604181088, -889130260747057664867163416)
(7060187998740983920, 18501933306410795273496111784)

Commentary

Specialization at T = 605/1052 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -25598589065575488253872T^8 - 219912513106757586234048T^7 - 324549648424331055742464T^6 + 1523547434916270832355136T^5 - 2849165156820063127691880T^4 + 3746564993881929295112904T^3 - 973581348748449447744444T^2 - 862126587120769476566112T - 118556909623229874930432, A6 = 1625048868702522016795790220948864T^12 + 18521876511799709835187838396715264T^11 + 69457772033476867393403462775818496T^10 + 74100738882710387630246485913752320T^9 - 324420440712638126863027033199077920T^8 - 1791949805594601002560679464033967520T^7 + 4689003940396889037978292829416265040T^6 - 2864400878761409044683373393741992480T^5 - 91393345783003551423720760724759820T^4 - 338626380479684457921000808774737600T^3 + 374448354510478802333765563563322320T^2 + 177989462843015858245005305484080640T + 17806843321561611911434592141844480. It is a 2-neighbour of the fibration given in curve#3900: its Mordell-Weil lattice (rank 17, no reducible fibres) is a root-free Kneser 2-neighbour of that fibration's, realised by the pencil |O + P - F| for a section P of height 12. Constructed and searched by Claude (Anthropic): a Mestre-Nagao sieve over the fibration, then a 2-covering point search seeded with its generic sections.

last edited by apkorte at 2026-10-16 04:26:21 UTC · history