curve #6166

y² + xy = x³ + x² − 8851988826450954918462586542794008x + 342091693600765887398192612490194091859453280864548

a-invariants: [1, 1, 0, -8851988826450954918462586542794008, 342091693600765887398192612490194091859453280864548]
c-invariants: [424895463669645836086204154054112409, -295567223271061729898754394713871466013098790072812477]
rank (lower bound): ≥ 24
torsion subgroup: trivial
conductor (N): 6367962895147526311451435040290000325942492681321729832833344746103158125918479216851612530
discriminant (Δ): -6163767456202031213404300398521772169572903269819683866487947358562336893472785944386225828883233519700
Faltings height: 18.4700
naive height: 246.2415
primes of bad reduction: 2, 3, 5, 7, 17, 23, 59, 73, 5209, 1009927, 2392073, 1624406390313821083, 116122131419203006865591, 1083685101454426928930881
regulator: 18513905380653232454953326.696138291228239518572308228992582379743884773171636364328077
submitted by: apkorte
submitted at: 2026-10-30 10:43:12 UTC
last updated: 2026-10-30 10:43:12 UTC

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

(9986474036124859, -15958931194589233515668312)
(51679220911679696, -4759166637482855084561854)
(948627888671543806/9, 650287239068189463887224606/27)
(48361278568748886, -5206361123527890870482312)
(42486936389473834, -6533962504157474098162922)
(90865548872231293, -16970168975843222471223755)
(47927615122051584, -5284828299739504853551162)
(140426900100186304, -43222826088692450452962842)
(16333034894400593941/16, 65738271103975334098186403077/64)
(161672600035506874, -56006882248499669783388032)
(31362088283411742064/961, -279409704576842961869563356062/29791)
(526738191934898473/9, 133614643469425891592941838/27)
(136911927633407184, 41189148218124929334336238)
(28038907782295074, -10767322500569317153547042)
(80196584467744093, 12164543858431456731349870)
(-121094173887769589/4, -193051692846983846508797071/8)
(674814360609844327109/841, 17414699986065269763938819411302/24389)
(58777178572653106, 4985741473974648244794766)
(-92019359081306826, -19428493533479233239600392)
(83881737972906574, 13775907860131209538148218)
(-34443750824005667, -24619591581507033619976978)
(188785613489468383, -73479822502515131428659980)
(175118904219736224, -64515485512213563049906442)
(634906172464358540107/11881, -6026757368966796140428888267664/1295029)

Commentary

Specialization at T = 9/166 of a rank-17 fibration of X1008 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -56986316681380998912T^8 + 24457666340476410624T^7 + 535487594425392365712T^6 + 48800690383254862104T^5 - 361470535143673852503T^4 - 1833035186758586060700T^3 - 8027564283553513223079T^2 - 1856193563269433892114T - 193818285257580953532, A6 = 176927718245814623549804322816T^12 - 116292625590383842700445954048T^11 - 3367104967138625608924904544000T^10 + 2020152346629037937923069338624T^9 + 29203681898463675958459856569392T^8 - 13720510267525899173180962213992T^7 - 142330533991632689827729911017922T^6 + 21328459230505455903018021874128T^5 + 318228745035755178053554583805228T^4 + 157407522251338495321687387801980T^3 + 99390588041592331039607736906018T^2 + 19564837728780267552757168309932T + 952356233459948553164561681844. It is a 2-neighbour of the fibration given in curve#3589: 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 2026-10-30 10:43:12 UTC · history