curve #4229

y² = x³ + x² − 323861019286741510104250926917580786625x + 2243651544708720480140891781019673534792677239743235171875

a-invariants: [0, 1, 0, -323861019286741510104250926917580786625, 2243651544708720480140891781019673534792677239743235171875]
c-invariants: [15545328925763592485004044492043877758016, -1938514934628334494935002472355579488970897402090418455048064]
rank (lower bound): ≥ 25
torsion subgroup: trivial
conductor (N): 2216987861263300051422436401674628016398716297779675144101794536149207740481628334022892920
discriminant (Δ): -693687100616409141695428967936605468522672942767392990235231206589050846243448668222856042634923056207416620000000
Faltings height: 20.8466
naive height: 277.6341
primes of bad reduction: 2, 3, 5, 11, 13, 17, 23, 31, 190168928111, 7939969942969604325803424687821, 7059097225133342226646758291845914582637
regulator: 26660264120503413718451386.24442109504208931334629285280744213370916622781815374222819576
submitted by: apkorte
submitted at: 2026-10-14 19:20:34 UTC
last updated: 2026-10-14 19:20:34 UTC

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

(4098836038201070543375/361, 37707438595070887259511871655250/6859)
(19522506756144596755, 57979753660372621767613951170)
(10415626951288492055, -615012044040656760272333130)
(10484520083962838750, -797963608950828367992998625)
(8781439708559042075, 8766680727451714220111651550)
(-8323009919515354645, -66049938418316282952259392930)
(10572062105689616225, 1181621217563192535771148650)
(15737710524278094125/4, -256792229536633246454706439875/8)
(10549471361402369375, 1074180674209142719886774250)
(6397011740293513175, -20825123677394052758014394550)
(12765629995810018163, 13771905430246644508961916354)
(10129954580890568975, -1565037881702546408414764650)
(-44909572364441741755/4, 534535594397099252510491616535/8)
(9272667356309689883, 6154824535088161466821418274)
(9064397760204306950, -7266865940092927740445088325)
(97194526113764587455005/8836, 2897988979614155666453667236245605/830584)
(56865526465954722275, -409527166899325338431744601150)
(12870716515068268550, -14402469997005773706011456325)
(46620850483880996525/4, -57840049742261978220461964675/8)
(9132868524453684275, 6901991863645021209469324350)
(35546226190928069375, -188800124058527623305922761750)
(-19857774939898914325, -29056325447190964033197779550)
(10404846374519993375, -603884805231246445383369750)
(10098745424903137475, 1725881790376505549811283650)
(10027552984876205615, 2099215440194452562486365290)

Commentary

Specialization at T = -1181/80 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -10669947921935414766592T^8 + 11909462495223346810112T^7 + 29290850738695015599056T^6 + 749826724411797003953776T^5 - 2800296645908250177432680T^4 + 5291011042236720751525024T^3 - 9053743075112346775820704T^2 - 1373686135035664689321352T - 153062929487822447776267, A6 = 424084459520579325699737587892224T^12 - 702971076900016054991835746922496T^11 - 1428460164091780099364248773693184T^10 - 45531551300070380060674898437602560T^9 + 178145138683306119166957105019549520T^8 - 224307849011465835658368095576866752T^7 + 1890363249575504635685922631864051728T^6 - 8402714629932209203714457518206796848T^5 + 10932033646871752694840038266436614840T^4 - 1045636958253350245306227479236779680T^3 + 3911537622199302231442442603656239344T^2 + 397823992391227625022128206594583384T + 16543056563137220502197778370669126. It is a 2-neighbour of the fibration given in curve#4205 (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 19:20:34 UTC · history