curve #4641

y² + xy + y = x³ − x² − 1364895602339402067224801462979816975512267x + 674263768238787148362828164994178832825249884443746595004234459

a-invariants: [1, -1, 1, -1364895602339402067224801462979816975512267, 674263768238787148362828164994178832825249884443746595004234459]
c-invariants: [65514988912291299226790470223031214824588801, -582563895758312096185188717104865200714495343043393417616947923201]
rank (lower bound): ≥ 27
torsion subgroup: trivial
conductor (N): 17984475627235815753345124804186459602778915872970873955731747764282717775024241574505607568113683745180806871490
discriminant (Δ): -33666872076214494848150880501968974223437450830006001925300081405222289423034574378110357110037076876694712228902828939264000000
Faltings height: 23.2056
naive height: 302.8606
primes of bad reduction: 2, 3, 5, 7, 47, 193, 12601, 1356589602179057, 184097786862976000802134949924953839102217146822749721562620206256708540975846273985109
regulator: 5268355214405083157346327513367.6953596682517183592751934914686310362475278430115751286803609
submitted by: apkorte
submitted at: 2026-10-19 13:55:13 UTC
last updated: 2026-10-19 13:55:13 UTC

Witness: 27 independent points log in to add more points →

(582086880247271891947, 8775056219712781792629006403466)
(613484532985643090317, 8234954362028494967157427851116)
(-7491713974162816456037/9, -948323791585196944450365342915898/27)
(1896004438968045703927, -70015960005845869826907005181094)
(1506675214170334213817, -45144965395912205453778376135184)
(-1197400530395043346253, -24326855254458216766634034105214)
(704254364636640582667, 7894433813096068630599278141066)
(-873406544584884024533, -34642516533610960512831583876534)
(99396606774961782001063/289, 76981401419835517363976679071203258/4913)
(6151723573022987052877/16, -919290225363486510014001974652301/64)
(195929796884381499580067, -86724744908777715377546599501978934)
(-1215714419422673395733, -23169114480050823300836008098934)
(-215989741022803609253, -30967579107267506266635933628534)
(204538494944149477867, 20090971793694572947200323453066)
(591019335175840905067, -8604034440674338552723593926134)
(1182552253000521829867, -26719276674644551467440509314934)
(782646476402550246859, -9242931929997855989526299221078)
(1390524463911052206667, -38275359434975073619388719247734)
(970077594787037733787, 16220393629534573705784918851466)
(-873315814117256885333, 34643725707150051528798455544266)
(1624363303696761433183/9, -562352441184418480772318613700918/27)
(741530154667481917367, 8360350509639402647031239455066)
(2997956902511075618521387/961, 4874867341059090182490200328658430006/29791)
(-762783167570703197344517/1849, -2716216068246326847313114077940545538/79507)
(6612639060452455571947, -529905217189876776128504652796534)
(89343978771954738561667, -26703058422829673271900730856102734)
(-606569625741356006933, 35763037002380422750228728221066)

Commentary

Specialization at T = -3364/729 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -34563528913036557884400T^8 - 363066894139066649206560T^7 - 1253243958943048847118720T^6 + 1217803890141503318620800T^5 - 1229632829129776891166640T^4 + 889782194202589151280720T^3 + 747172710296992900801080T^2 - 433583331227756476888680T - 198981396396273976948827, A6 = 2529174685018796943057183053750400T^12 + 38255811556046455632742522418874240T^11 + 209255178696584738819667007675000896T^10 + 459298877624289867341928240423086400T^9 + 710742840439999694736111158271090960T^8 - 2594464885975235791000216537446763680T^7 + 507169769440286090575053053081870400T^6 + 1519561014747819598698469788704128704T^5 - 146641249872692903802186384657680160T^4 - 569600267540526989066094026512214160T^3 - 74471423161824441814957627679250840T^2 + 111793067512606356283231367322794040T + 33201984796393433668897397905886454. It is a 2-neighbour of another fibration built here: 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 2026-10-19 13:55:13 UTC · history