curve #4549

y² = x³ − 4662207840217014752473318532088092957971095844923577499475x + 122437964858932935093951334021255900810328537431641118383210183405054609531545203001250

a-invariants: [0, 0, 0, -4662207840217014752473318532088092957971095844923577499475, 122437964858932935093951334021255900810328537431641118383210183405054609531545203001250]
c-invariants: [223785976330416708118719289540228461982612600556331719974800, -105786401638118055921173952594365098300123856340937926283093598461967182635255055393080000]
rank (lower bound): ≥ 15
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 643282095696179055572916818493972668921615787215139585868671202490355824559600 ☆ record for ℤ/2ℤ × ℤ/2ℤ torsion, rank ≥ 15
discriminant (Δ): 9534401296453017120398739624809385523393692790911683892529118401746568318018284899910088812214575279969736133856482372451347589624589701142854590205861091414614050064000000
Faltings height: 31.9268
naive height: 409.9741
primes of bad reduction: 2, 3, 5, 7, 11, 13, 19, 29, 31, 41, 47, 53, 61, 71, 73, 79, 83, 101, 109, 113, 157, 181, 193, 227, 233, 239, 467, 643, 751, 1091, 2039, 2357, 3217, 3323, 5881, 7699
regulator: 19729028045092635.833988058855598983360677549493993813752660893144004
submitted by: wgxli
submitted at: 2026-10-17 15:01:05 UTC
last updated: 2026-10-17 15:01:05 UTC

Witness: 15 independent points log in to add more points →

(40312723703878191617647113465, 67143441857180609573327416726342329671700)
(40322466540655701703378089531, 81217629274080099343545360230625585186354)
(-78836832708874342957077084915, 36608237939324169644154072750939112880400)
(37510014337541590191799555225, 578892215860937453431495880319581564670100)
(38162061650212192856242721589, 309113657279997568598462911539273007937712)
(40346121814484875082588146485, 108451027540226733513084049427632060225200)
(40429855764057440507622681294, 176449847671655955628000263261586533609972)
(40923997921015532816744769465, 424514383841157433621894704696691117214700)
(41314728382044113980391704425, 583527481674018063750755478630229188179700)
(9879611851783578530877196585, 8794400356769431270419195866721156360293900)
(24805282135239169494387608475, 4696096704559711195510804435881834068427150)
(43354067837485962729656090190, 1341462084421087044317999463265179809883900)
(-5004331715425800765749781015, 12068300388101183401008846342993946882067700)
(-76220017023904991833362982359, 5915405587164343929367764577329986959962636)
(1808074795756116341293976451435/49, 275726256771277602349303688295681660426431550/343)

Commentary

specialization at T = 4206111/3370112 of the rank-5 fibration of X400 given by: y^2 = x(x + (286 - 275T)(64T + 19)(T + 7)T)(x + (23 - 36T)(75T - 4)(89T - 81)).

last edited by wgxli at 2026-10-17 15:01:06 UTC · history