curve #4206

y² + xy = x³ − 5665984534768836476971609755818855850x + 5219683802894087570340363768228062076043107401635783332

a-invariants: [1, 0, 0, -5665984534768836476971609755818855850, 5219683802894087570340363768228062076043107401635783332]
c-invariants: [271967257668904150894637268279305080801, -4509806805700491661182025182252401860043200697432274420049]
rank (lower bound): ≥ 23
torsion subgroup: trivial
conductor (N): 723978752176623467066621027541941900700214159357362242896740061914841236010230
discriminant (Δ): -128458203615624650995216763709151639278354750581132539799094319899995823280281675468797466643613214082560000000
Faltings height: 19.9651
naive height: 265.5072
primes of bad reduction: 2, 3, 5, 7, 13, 17, 59, 103, 13993487, 183442122610296247039350106173500616764948750117248806629235397
regulator: 10455661477994131633140.2451443233221112054969299275762773977703428414390824336950156
submitted by: apkorte
submitted at: 2026-10-13 21:11:59 UTC
last updated: 2026-10-13 21:11:59 UTC

Witness: 23 independent points log in to add more points →

(535037009788386324, -1530141999064567770617117082)
(871499077260724404, -971440339546385672157878202)
(1292929246504717404, 235186301219037676665564798)
(1362038657239825524, 170818329612928103707562118)
(6410478749958461124, 15242432497262818646246761158)
(1593871716832107204, 487797219508385264889582198)
(1546286337888240804, -394489778447416176370147002)
(1725132876083074404, 761082774624188303305971798)
(-359727259528087596, 2685394357307401282322277798)
(1564457609368898904, 429586287266292359412347298)
(1991968440935797452, -1355444294058806350373658930)
(1797953579871143004, -919041908104639108713097602)
(1406068099505663904, 180994464648238591876706298)
(1973566236105780444, -1313185767303671429344092162)
(365243040499058304, -1788559954699243754643007002)
(2584999796402003724, -2801189476789864920022269042)
(1297357260502852404, -229142225016502183040702202)
(1941630515534440404, -1240259905874907195231182202)
(1808332337875175604, 941842106791244424822427398)
(1346625314710326324, 178032549135765467262850758)
(1423586776796530404, 196731091817191196559561798)
(-520942407659263098, -2833718680339335974348124480)
(1345399794628820604, 178824938534227808941816998)

Commentary

Specialization at T = -186/163 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -32314259208676675198192T^8 - 220263400022708733054624T^7 - 255006976139783552896608T^6 + 705363094453509328737024T^5 + 1047835198424784109021200T^4 + 1820453244371835449110416T^3 - 1245666920662047666014424T^2 - 3926333297851033484227032T - 1657966129723797525563067, A6 = 2220158269397315819350729045585024T^12 + 23424072793928593017661886421060992T^11 + 69989708296427149663267141905289920T^10 - 61968539237944624284964605273911360T^9 - 599227600313137750772024638621900656T^8 - 203890468677123352982190860910686112T^7 + 2258581157822950043811972713225053920T^6 + 559769644036858048987416888208533696T^5 - 3908030961820075418207725091990624736T^4 - 1985850721280648599946097200370542160T^3 + 2921751862616849299905524795576315928T^2 + 3089933922534003123866046309582827784T + 831153037936820603440551361962407190. It is a 2-neighbour of the fibration given in curve#4187 (itself a 2-neighbour of the fibration 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 2026-10-13 21:11:59 UTC · history