curve #4115

y² + xy + y = x³ − x² − 1249826097099625684027737153603030292160610958707272751440x + 7785143862885830951791316765517692588525403518394334275418164717920828757438258529587

a-invariants: [1, -1, 1, -1249826097099625684027737153603030292160610958707272751440, 7785143862885830951791316765517692588525403518394334275418164717920828757438258529587]
c-invariants: [59991652660782032833331383372945454023709326017949092069105, -6726364297533357942347697685137323959512429492142713588783039773176904079345884455252425]
rank (lower bound): ≥ 21
torsion subgroup: ℤ/2ℤ
conductor (N): 2354821899258314703620981481029366760380391230269986226723630380510794423225946002025617184809098350
discriminant (Δ): 98764979522445525354120949581381016413251963321297130105295511726385275781802955903383532930732474241183846612083181058902381033816188232188172227912347481626452216029184000
Faltings height: 31.8895
naive height: 406.0247
primes of bad reduction: 2, 3, 5, 7, 13, 23, 29, 31, 43, 61, 71, 89, 157, 229, 359, 577, 719, 751, 1543, 1951, 2063, 3331, 3547, 3583, 3659, 3931, 4421, 7877, 13151, 99079, 114769, 467827, 1516909, 2985617
regulator: 17743173740698309216325823.402371547024670638149098713623550246797259747666987801
submitted by: wgxli
submitted at: 2026-10-09 15:30:08 UTC
last updated: 2026-10-09 15:30:08 UTC

Witness: 21 independent points log in to add more points →

(191279894202447109878450833, 2747013698147413501625695351209437661422181)
(526293222494255363012108093, 2669740555518537770259028311552787598251281)
(6161243665563705410911011275, 564399892423454204327903574017009800533003)
(32163233371152608225794375205, 926666405391619358464159958919493900187961)
(5454920956949794586683959629, 1062901128277456401233397219975590107881345)
(-1398817922156758999072167751, 3087180906885552383019256360215198916038105)
(-3137918511543314933377256827, 3417030683548427433277806465863302202885145)
(-38118183326034409187406532387, 201693001052595454663855338783665912721105)
(-4853600656143521167030248771, 3706340813369441751642057049837960858313745)
(-8125033626331732915076475171, 4171766929748687920261626298754997027800145)
(33243230968374422334349168073, 1724642913572288763046515549175366259354745)
(-28923650258700114323629961251, 4442724442764243926748439790733524296904145)
(-33245116364974996021170416611, 3548520802288944906685691705334869823671505)
(-37207204456009611572579511111, 1667008025185848202743595453368195869271405)
(95087942201771926460611863389, 27362382793070230300435501839251691355559505)
(137510923520056287138068749245, 49357364147982975916701685435783034487352881)
(151491059966988927185959593485, 57402944117402591335088978796192011057243313)
(259811849413974313073226858333, 131228564653906218522650419995287541325209681)
(-1867749282542509191529274100379/49, 71395078256775630156868023002815012248723735/343)
(-1549361648493111001353483420315/121, 6198700774470214283649578672853654647295214531/1331)
(374204344222382462546269825370339/676, 7224133242321136935441139273893288459890532521505/17576)

Commentary

specialization at T = 92968/9807 of the rank-9 fibration of X1152 given by: y^2 + (6573T^2 + 312188T - 7905)xy = x(x + (16T + 3)(2410T + 9817)(7953T + 478)(8 - 55T))(x - 529(349T - 78)(6T + 13)(25T + 22)(292T + 967)).

last edited by wgxli at 2026-10-09 15:30:08 UTC · history