curve #4219

y² + xy = x³ − 1993091370678968182622630865788394662660705x + 1074544813530136200792412413600165186884805897963334224207566025

a-invariants: [1, 0, 0, -1993091370678968182622630865788394662660705, 1074544813530136200792412413600165186884805897963334224207566025]
c-invariants: [95668385792590472765886281557842943807713841, -928406718890037677484787827929231607177621125262657534131048616361]
rank (lower bound): ≥ 25
torsion subgroup: trivial
conductor (N): 734643589999131929201004058305667710915325726599822550780582597629519219231224255521423999590
discriminant (Δ): 7905167260073296847962111912157683330393727730919115999472489596859694727963723173690587085168657407408922402173672038400000000
Faltings height: 23.1720
naive height: 303.8084
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 23, 31, 61, 97, 1804377679093114436633352127, 189040081051771170380233813120652878800344527779412927
regulator: 21397380928977207070892410.21251332988081709401098785835724218548322895475468470199779126
submitted by: apkorte
submitted at: 2026-10-14 07:58:43 UTC
last updated: 2026-10-14 07:58:43 UTC

Witness: 25 independent points log in to add more points →

(-615540265478228695190, 45476926738379105426501930187595)
(884094723459331240810, 1868808117971729687160764427595)
(1243738065294975958122, 22794301657523004785716711489611)
(-1476975046347220786490, -28219426086718210404292878774605)
(17844255438221190828810, -2376432550143095409906944665632405)
(1603693399204012504810, 44751180680604391111082662587595)
(-874788002862411910550, -46353448319712522602910107025845)
(884958435665961621610, 1948912235872318329793087486795)
(874928224092729732340, 699963115366265036561991818395)
(433989584204248668010, 17067645906482394528840916683595)
(734373801597633177010, 2630988788333666568031660769395)
(697404222704450287210, -4873685996940537032175211220405)
(519368699042614354810, 13397464364429295174860879037595)
(683341248522050325930, -5627882334237520480640440236405)
(2566010740165475610730, 113383952118687306401891879019595)
(2942855294239065738611994010/151321, 159234774843691012994318467982467089865055/58863869)
(1997580829239504904810, -71163113701695495060940954292405)
(-1628743254481649911190, 203369776477379903641191243595)
(1185982901485175037802, -19465950574345146765194175242933)
(-7833256448529655190, 33017521541060448096556407627595)
(900709487408168953510, 3173944005460458114568170785395)
(753405630304596664810, 765956919215445987602538267595)
(2161948743504920128810, 82888963184294589780231384003595)
(49081030975868767088039290/56169, 5222954897192390864553297979249446535/13312053)
(-424368597200876779190, -42940961923552783648415794172405)

Commentary

Specialization at T = 499/1004 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -10490924381115083744475T^8 - 198606534222460688122680T^7 - 1065417976804438457255016T^6 + 1639325103621379811930736T^5 - 2065857830370934339051440T^4 + 1624515338975555277541440T^3 + 627292956712495652706816T^2 - 579571543143419272734816T - 175069545700351062329712, A6 = 377475523351270352670804404613750T^12 + 8093831427752091624508772362041000T^11 + 59608190005431403688311326659353800T^10 + 258877237945365511005499173975059280T^9 + 1166329210851912140109373033476192096T^8 - 3290737074386209591354797503931856128T^7 + 1076110014915408676877000781319873728T^6 + 1655411108858774108267005356846449184T^5 - 323687792595432052178005121163318384T^4 - 745457216313233319000016951349121600T^3 + 632802115601259765911745996693312T^2 + 143229663791151165669809219876399232T + 28115415195462890182082065162444416. It is a 2-neighbour of another fibration built here (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 2026-10-14 07:58:43 UTC · history