curve #4217

y² + xy + y = x³ + x² − 4300997521895559353296539778416519698724040x + 3414834408214511764576735208770865309374074165422095836380919881

a-invariants: [1, 1, 1, -4300997521895559353296539778416519698724040, 3414834408214511764576735208770865309374074165422095836380919881]
c-invariants: [206447881050986848958233909363992945538753921, -2950416928697338164595847579485910028666386833244920749724655431745]
rank (lower bound): ≥ 25
torsion subgroup: trivial
conductor (N): 181067958205627543116788725873078333477524043431875412351138736291524758324425666817398010082
discriminant (Δ): 54397480760610627491991106755775836752382704750033260516000923468729308610097682849464633956735419009640455785470770124460326912
Faltings height: 23.3483
naive height: 306.1159
primes of bad reduction: 2, 3, 7, 13, 17, 19, 23, 31, 43, 53, 61, 71, 157, 86399, 10262801279, 1047973045800633191898460783061214030886355968884601207808011
regulator: 222159866642113008061909747.4329240233444382915770442679225668901379276595025353719749910
submitted by: apkorte
submitted at: 2026-10-14 05:17:07 UTC
last updated: 2026-10-14 05:17:07 UTC

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

(435664465119546396281, -40295573889456964644901846477923)
(1045039056991528360309, -7836969901302937881747116634371)
(1972360126392174820261, -51035307082167995866787277328179)
(-988682944646254991131, 81857960464668594728700383877261)
(1276139463301652048101, 2096569058617912808400219396237)
(410320290322361575661/25, 7228683498125580146029946542295001/125)
(1122740432909793891377, 1093910302874945358642456434589)
(719199446186191227749, 26335598244779775759080414840589)
(1759172274798737680597, -35954506615232459249207095379571)
(-825696971775346138011, 80020091732394090961361663730061)
(-21533235148842876729707/9, 81515834221886335450369854871303/27)
(-886727747455490054203, -80817241674747377295548815341171)
(605223477800830992287, 32147477988534373070584504764009)
(887975884236536829801, -17199487060751468181679086773639)
(-882815553721409594011, 80769968875951263382469770610061)
(245574901714771577509, 48717833059673084057584545772429)
(1292219801940407585429, -3845808107481182523074654042739)
(982383258775803238629, 11733789002277734341573193605261)
(1316196435321650153819, -5832670314328121450533846531989)
(422419361292044215108781/841, -906417850834564774977419668203930111/24389)
(63101036091645440851237, 15842474067371249214906651732480909)
(28019392869275425216649/25, -186665999457663096240333686101203/125)
(589420538844167559557, -32931938815583985126155549189907)
(31744128394505724381, 57256746143000593057544400694741)
(2204745406662658987269, 68185643018435627364099483301261)

Commentary

Specialization at T = -63/683 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 05:17:07 UTC · history