curve #4076

y² + xy = x³ − 9816912725444562187524600797902415008202105152126113863x + 11818799450661452554897134080088006676176269596921055787278149418755224711667070217

a-invariants: [1, 0, 0, -9816912725444562187524600797902415008202105152126113863, 11818799450661452554897134080088006676176269596921055787278149418755224711667070217]
c-invariants: [471211810821338985001180838299315920393701047302053465425, -10211442725371495007431123845902855484448305409241563457657294978395065721833428865625]
rank (lower bound): ≥ 21
torsion subgroup: ℤ/2ℤ
conductor (N): 36002324741262229107384053035934129095312333739830493609470477957959545215672356665910453744176850
discriminant (Δ): 205194931593499080300675582154453277266288091437637331516748908813508343496007999587380988696481051980236176662342201627615903877404833910853213181972238372864000000
Faltings height: 30.4168
naive height: 391.4847
primes of bad reduction: 2, 3, 5, 7, 13, 17, 37, 43, 53, 59, 73, 211, 241, 283, 331, 337, 347, 397, 457, 619, 967, 2861, 4603, 5413, 5641, 6607, 7487, 19087, 70051, 10951669, 25632109, 12957651503
regulator: 672735844734286815385534.77350783131017871343439993342362289574721168368987165780
submitted by: wgxli
submitted at: 2026-10-09 04:59:18 UTC
last updated: 2026-10-09 04:59:18 UTC

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

(1869897417259948620991895302, 553709463041026883056968832624507047349)
(1696965298890279827328971152, 6824609496062322097493151544133690686099)
(1744897658693291519236235062, 1387517478609645368162848745620016490469)
(1926129114353817515108389342, 7486370898292665355827763540834505169229)
(1926743210681463891444921658, 7540175168721055487804954314534546262617)
(1930713891470445761027216242, 7885883241135290689233565324083376685329)
(1955292075047400227986452342, 9963740608937193660252887178031686032229)
(261868637349197478945870262, 96260145346144799762049937107658472548069)
(598894616579718327071458902, 78449420978651560301550760277687905721349)
(602711195776124031007943542, 78236676728338304792757998084462237328229)
(623145611592757396897461718, 77093497269796930544351760561208991103877)
(880726230364795706784130612, 62096275921272152565563522678492801946919)
(905454435169859836309146262, 60600058174883423713432922883066084505669)
(1292366056513492196479180402, 35920431607153238174805176933027477599249)
(2002162483092525072614823538, 13773995687681822311532644537985634467197)
(7173118156016435811964394902, 557210776660029310944556863648683401197349)
(8475036388088123478580443862, 733041690208686838016547459501982954561669)
(13877420898739704655340323342, 1596288850107710156307070708892320348698229)
(49478928951635917680181143046/25, 1489912678370525470283419649452539600925841/125)
(453013093513374335546189523142/49, 289275828049811647676184911633571859669999075/343)
(327689495664949077185270113942/121, 95008074523507178837076717621467062009968719/1331)

Commentary

specialization at T = -83339/23895 of the rank-9 fibration of X1008 given by: y^2 + (7920T^2 + 62574T - 34198)xy = x(x + (-50T - 56)(512T + 71)(6241T - 764)(T + 61))(x - 2(625T - 67)(T + 8)(32T + 67)(6889T + 17269)).

last edited by wgxli at 2026-10-09 04:59:18 UTC · history