curve #4275

y² + xy = x³ − 1197304805159734487179984550530066246594144975x + 15896542823033510196201677887690482405517585258649841003511904830457

a-invariants: [1, 0, 0, -1197304805159734487179984550530066246594144975, 15896542823033510196201677887690482405517585258649841003511904830457]
c-invariants: [57470630647667255384639258425443179836518958801, -13734612999100952809518335900910548299250270622361100791804040551953049]
rank (lower bound): ≥ 26
torsion subgroup: trivial
conductor (N): 366606529437414781286685203550676921309473368465239452431786719081938621846001976326937135318223990
discriminant (Δ): 682073071496910640466714209685740303862015058694647589172412713657280723783631477304716361147370697213239454105692985762839763353600000
Faltings height: 24.7335
naive height: 323.0028
primes of bad reduction: 2, 3, 5, 7, 11, 13, 19, 23, 29, 89, 103, 149, 111697, 1615515320713926448564207, 10367116736264996759085139193022296339940641734673489
regulator: 719225472036500953171648023.971609363868281735443665608596403207238178342290054082600671429
submitted by: apkorte
submitted at: 2026-10-16 04:40:41 UTC
last updated: 2026-10-16 04:40:41 UTC

Witness: 26 independent points log in to add more points →

(30661225339747830803054, -2830310515617653731816292612560627)
(9397766098888623551174, 2339774410761436690381560416628893)
(16505010438998747040454, 794501752072535124944732273980573)
(76688914640409893203604, -19367462049642845027522787961208227)
(18876785638346053744454, 147305836213727590093487104642973)
(530675711255784904436054/25, 26791189732690690205293093876466857/125)
(-22214522400445898184506, -5615297278753237778003900927621987)
(-37779798369101820156046, -2684562044754718392974270089416427)
(18404149863770689994054, -308035128828395619195589499542627)
(-896961626855358165946, -4119436569150785998345320379222627)
(15023504861152953101878/9, -100672566914940829762929057965815729/27)
(18185014136443311051974, 370440982778492149096978110578333)
(17513870350102229078054, 547019362567679133669779662532573)
(20211336497515302890699078/961, -4016002126690286094078289382762732989/29791)
(24272554428556624365254, 1065476258621863328908392479266973)
(55992900861215316920924, -11153707604938988824854431117738857)
(11259810788037638001254, -1960273876111769500542514772005027)
(13984759886553828108998, 1373889069960413798848281953342621)
(29031921188858918767334, -2367722518032937764527020630667587)
(37793198574902366731214, -4962616987832565571661947332344347)
(19038853154123618001782, 48954333715298638145367428587325)
(21303608062133570330054, 241120675241006140158380656585373)
(18947016161791544481734, -113875804841252069971310760652387)
(3185837787751513089687494, -5686040883792595499460639393368389987)
(103861784268668758202054, 31810807331334944654090280821641373)
(26865820478120535420852422, -139251474515920052188232752113876680035)

Commentary

Specialization at T = -14911/6170 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 04:40:41 UTC · history