curve #5966

y² = x³ − 87268883821750281046090916390901975x + 9940673266113788666408338901575131718114881753742650

a-invariants: [0, 0, 0, -87268883821750281046090916390901975, 9940673266113788666408338901575131718114881753742650]
c-invariants: [4188906423444013490212363986763294800, -8588741701922313407776804810960913804451257835233649600]
rank (lower bound): ≥ 24
torsion subgroup: trivial
conductor (N): 2314167259789826544946546589393075654953902813678227588751359933456947430542982916600
discriminant (Δ): -152781718279024499091239295278486132762610091935151062601391863096716125136679737072290785960688298720000
Faltings height: 18.8768
naive height: 252.9801
primes of bad reduction: 2, 3, 5, 7, 11, 13, 19, 23, 113, 131, 1663, 25541251, 16390347970059334554573498893, 285190068998030142196440106061713
regulator: 684350007777956656417981.62606386084524543908926817983805758377593066606672098825608070
submitted by: apkorte
submitted at: 2026-10-27 01:58:58 UTC
last updated: 2026-10-27 01:58:58 UTC

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

(179318780814049119, 7599946537097772589214028)
(183193278900883165, 10076009139988022124409670)
(153673281407938899, 12603389677440732789887268)
(160897650082059315, -8040022129210241539949580)
(167546078479842655, 4734481865616832715820980)
(-41318906409012626, 116086115976245066460740932)
(133987598751982725, -25557016814674915789263330)
(-574041448203354615/4, 1117397958089326028869625235/8)
(159449435772634705, 8919696112746160657372730)
(171857110695469806, 4321094742122587586961804)
(356665193758997925, -155519454613622889943528830)
(174672656661433845, 5151896394146996044326930)
(156002504114722495, -11095390122310394191422820)
(-126982638974824845, 137748914214530347631104020)
(187013395924133385, 12681930885593059563643170)
(-224752008071852655, -134913036523838838447111930)
(116217419198615250, -36989161261632967708445670)
(194788673250975795, -18233956991309321728439820)
(90974719179658206, 52481941564233858410437704)
(97861292029406655, -48349030405651203820285980)
(4653662385593971, 97645553541478262721931244)
(708046073926209465/4, -50209979956772440696693515/8)
(429672365290883230, 227528119150619649768340480)
(-8518753334552855, -103360907321932585810555070)

Commentary

Specialization at T = 409/34 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -96196961581357063033152T^8 - 399556103168445237840384T^7 + 121419290054525411589264T^6 + 1430959091691248074015632T^5 + 100688238281333139439320T^4 + 387367434659202151100832T^3 - 1953294802285379458973616T^2 - 733254399190976661626184T - 79743145896962463440187, A6 = 11484504086394253195526049377853696T^12 + 72121229539722792963703791386638848T^11 + 54055657239595166572471510008621696T^10 - 398904635104645281362553198961699200T^9 - 498516576461076920592919777700932080T^8 + 1133982302557837972386711623027647680T^7 + 1070918296969515869033113344560286480T^6 - 2589863488770145152071246194277744400T^5 + 73956315674627679114868875249120120T^4 + 1069188646911222614852645794843754400T^3 + 580961502887065292988342169113394080T^2 + 118245525237113913102523416049897560T + 8723796687794086545932578364805270. It is a 2-neighbour of the fibration given in curve#4191 (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 2026-10-27 01:58:58 UTC · history