curve #4585

y² = x³ + x² − 202739875286850067249669151085179625x + 50653251212239947253741263422690883658440567389032059

a-invariants: [0, 1, 0, -202739875286850067249669151085179625, 50653251212239947253741263422690883658440567389032059]
c-invariants: [9731514013768803227984119252088622016, -43764409047375314485621535679817742848797365736655431040]
rank (lower bound): ≥ 24
torsion subgroup: trivial
conductor (N): 152644332038686018241286373503957697870476328219471932265402133554019543264948099391096
discriminant (Δ): -575072978663020819214862706492392574346170950939057945877622025819877039800639910384445944826854900491725568
Faltings height: 19.3789
naive height: 256.2368
primes of bad reduction: 2, 3, 7, 13, 17, 23, 31, 43, 234948323, 214203636247232353584559, 204964478153866583233943714913921515910005053
regulator: 1821668140328206901342234.0819031584448730261418047308717423051803085785352475222578409
submitted by: apkorte
submitted at: 2026-10-18 10:32:32 UTC
last updated: 2026-10-18 10:32:32 UTC

Witness: 24 independent points log in to add more points →

(-273290436664687386, 292657985174162036709396207)
(2189566035786345363, 3178672228478801143503569610)
(-423273274376779225, 246239456933375630473608678)
(762406063798857807, 582444487865993108853052374)
(-43750707950698117509/121, -368587691269634387587441073214/1331)
(562941385044999597, -338999115701078644286616246)
(1753448321914662429, 2255277872691336947852048742)
(-291694145105389803, -291500173463521843909064046)
(-316974252548749194, 288217487359062855097767081)
(194412819601054971, 136330900028127217093864806)
(1842915337669203021/4, -1876925866404121918090629651/8)
(12701498228879923701, 45239181984409324722780448206)
(8577173537207041095, 25086179735519878248617584422)
(-2669985197145072639/16, -18083635193590105048901032041/64)
(5053689075415110291, -11317956623278079763958813494)
(376633915138015227, -166496933688738734287037286)
(154131482943073200789/4, 1913407954906620737174342132673/8)
(-109634496488997333, 267512166262030866144007494)
(494136587933171694, 266694337715873279527742427)
(295443851612884485, -128621264097413546293056222)
(143715815734817277/4, 1666908993608547806196772875/8)
(1706748473974132827, -2162491414744059369238909914)
(438118717670235, 224865353405442754691146278)
(1289620670389782342759/2809, -34705432981195522881441294577662/148877)

Commentary

Specialization at T = -139/93 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -94521237307405223348592T^8 + 194603007768925915807776T^7 + 374678018163025368194124T^6 - 569211691896707346596088T^5 + 55463370301014723924120T^4 - 1135612908646444410668928T^3 - 2702507543729482509654336T^2 + 609092474805433024526016T - 184694377125002807554992, A6 = 11265741534554236438342976060251776T^12 - 34833166213267258548363680994916992T^11 - 69502033189823387959197312852122064T^10 + 265587490260133017697161120524802480T^9 + 273144084685427886803017231992231060T^8 - 1114854493674664679669934784078225824T^7 - 766273999597311933358466543970856272T^6 + 2407762999977059159006184076291662816T^5 + 1357060199643920217443950358505612000T^4 - 540509467373796107556447739286515200T^3 + 808090270114106954000628418781159424T^2 - 154315127645781466989761857773289728T + 30433563297067405810172392059479424. It is a 2-neighbour of the fibration given 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 2026-10-18 10:32:32 UTC · history