curve #4554

y² + xy + y = x³ − x² − 26559208023520490753857265433037187x + 1687528652180557796587015519353707019252407022762499

a-invariants: [1, -1, 1, -26559208023520490753857265433037187, 1687528652180557796587015519353707019252407022762499]
c-invariants: [1274841985128983556185148740785784961, -1458024755484001930514392475641176861800910334130767041]
rank (lower bound): ≥ 22
torsion subgroup: trivial
conductor (N): 663374512441231775244000065310658129112072096871191143483321476920768010
discriminant (Δ): -31212289280637684803783467392949391351864425222487670136101082813307760047293555078566390610521600000000
Faltings height: 18.6618
naive height: 249.4334
primes of bad reduction: 2, 3, 5, 7, 17, 19, 23, 41, 59, 67, 109, 266647, 224680467559, 1636226654825959639, 81847462680561325593697
regulator: 460489485447418278549.8779311162795245517949740057298875629204226729167902377395099
submitted by: apkorte
submitted at: 2026-10-17 23:03:22 UTC
last updated: 2026-10-17 23:03:22 UTC

Witness: 22 independent points log in to add more points →

(-59544629793672683, -55297989346375745204580754)
(288543243977568907, 134340433961982394661302946)
(935533600459095307, 891987180800378437258113746)
(127398125419457107, 19278041462893566547660946)
(83916418721613259, 7050641213189512290190178)
(64278479073302707, -15681964743241711422796654)
(145295050884374347, -29931261324182508182331694)
(97700431338147547, 5026903280304618528427346)
(473705470582287067, -308875885945416546701678254)
(-86866372166027813, -57785442153359578401161134)
(308630058474450907, -151288953425878586583143854)
(74306595474407707, -11148361930463729350876654)
(2444454845375737, 40281763539300446874626696)
(94972202069334427, -4665273051463630800830254)
(108759881520267227, -9243464299667470908062254)
(-33956189427636443, -50499761272050876685462804)
(229765186325624107, 87834619334247100409746946)
(1479763827038833237/16, -302011109310757259725653931/64)
(17011621512904095523/169, 12812784717019352505936699962/2197)
(92082088811689387, -4761886195072564354974334)
(521101679771726107, 359654080871620856773876946)
(42783635761643227, -25090676105646812860940974)

Commentary

Specialization at T = -93/8 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -149283522995912209072T^8 + 7771051548175580374080T^7 - 96425969122530373515360T^6 - 650745155405088927481120T^5 - 2415911230346916965685840T^4 - 7601154786891989779697040T^3 - 2330949979661939283434680T^2 + 13731963511608379552174620T - 24775768582367122920991275, A6 = 728648747539615441339595057536T^12 - 56522770035420299634338505890560T^11 + 1220976136746860242842754846581120T^10 - 4534650284415406223365909009522560T^9 + 40082563358095594076415076965738240T^8 + 962506828193107880044950195752958528T^7 + 3523430440764385275475987785896930400T^6 + 2285241161704333099311601789013085360T^5 - 1404773498516823569998513341965901420T^4 + 18727747445156873508749884432613320280T^3 + 13182091133577376201427412811936218796T^2 - 39167275224385873529615083153972312500T + 47401684956970697155653713375708163750. 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 2026-10-17 23:03:22 UTC · history