curve #4189

y² + xy + y = x³ − x² − 284037061454951436351661150009232405867x + 1800036814468770187736885056758932484284645523256867145659

a-invariants: [1, -1, 1, -284037061454951436351661150009232405867, 1800036814468770187736885056758932484284645523256867145659]
c-invariants: [13633778949837668944879735200443155481601, -1555231807701017442143316683765448156169974923691939014182401]
rank (lower bound): ≥ 24
torsion subgroup: trivial
conductor (N): 1995026868259792220925033067480146082178558945648195128203283132535465250851319644490
discriminant (Δ): 66840208365375964394195629531099444292324786075131242827274711064622671996299031584236653908332104679478389760000000
Faltings height: 21.0060
naive height: 277.2401
primes of bad reduction: 2, 3, 5, 7, 13, 19, 23, 29, 41, 47, 79, 97, 191, 23539, 587813, 54727019917837527982984475073613251020888263874071025419
regulator: 194768232689864400108037.63917393864080176301348664817520202139802040096169537524027373
submitted by: apkorte
submitted at: 2026-10-12 01:16:57 UTC
last updated: 2026-10-12 01:16:57 UTC

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

(739375616386560907, 39880207323649106504237657066)
(7175505684049815467, 11462061512134208429219737066)
(-14001956220734132933, 55063244512771185094524127466)
(7296583605441639227, 10770688233052076256849529066)
(36734395430000143867, -202326668943544915264203501334)
(16480896968989497707, -39942357544231653218721958934)
(-5960865438087764213, -57283003956330218781849818134)
(33787571509488499267, 175428063690675284698353588266)
(11654208617855361367, -8525901811850660550559830034)
(7565044735185435307, -9177717898871917930282230934)
(6210115805430936607, 16602115188813173596185472826)
(5130192334071846067, 21860762602596020925640231466)
(-13104337328516266933, 57199908706875114395596479466)
(7894627535024397067, 7050092579043924784603327466)
(-17805293882106793183, 34822618083520051042982174966)
(6660780018389380987, -14270267476489008809383640854)
(-16384786512853617413, -45334793504372164514929436134)
(23362999825881513643/4, -147577951144037864056558329647/8)
(11372551563708637867, 6377996301639873811611554666)
(684666720697474263251/25, -15084149873907708438435486015166/125)
(13036184404683243067, 17682587679204656176644319466)
(3059579455966809067, 30978115833897753076952767466)
(11426173338393025567, 6808232022635594427242737466)
(1638566529432925478683267/121801, 864509232660199220092548665915546834/42508549)

Commentary

Specialization at T = -609/8 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -159593418515992752T^8 + 66594122607180400704T^7 - 9577916877180584429856T^6 + 378921666915696440595168T^5 - 8319401481873790592668080T^4 + 116272680309493432400708928T^3 - 728840755857976029224439616T^2 - 494050735965340585862790156T - 1390993922871848594371288527, A6 = 40790046358717726655041920T^12 - 19761746514541051225237666560T^11 + 3565895854811577447633633290880T^10 - 339557681946125505951450862243200T^9 + 23467317247680868700633154198453120T^8 - 886973909798626440691615033339785600T^7 + 17462193623232977878169289412672255680T^6 - 168744348646923648755279986373234058480T^5 + 648041970284178877886227382317646179620T^4 - 1598469487562293877807341226375296120200T^3 + 21038628236849997090209222042624669117476T^2 + 9643147208228838672635695562458112132532T + 18608807243853177902952156937051966196946. 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 2026-10-12 01:16:57 UTC · history