curve #5548

y² + xy = x³ − 1598946474985198508911736535590x + 778762161780257579980855045697058255810663396

a-invariants: [1, 0, 0, -1598946474985198508911736535590, 778762161780257579980855045697058255810663396]
c-invariants: [76749430799289528427763353708321, -672850507778142664227604958416550974665443736625]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 8677551978688959744721052117666650968801190477243895647930960878
discriminant (Δ): -368745720362423219890348942398080231927373545138942663009880667226992068827076356481957888
Faltings height: 16.1154
naive height: 220.2557
primes of bad reduction: 2, 3, 7, 13, 19, 29, 41, 47, 79, 89, 94529, 125887, 470831, 6244347101, 4680721312117450811129881
regulator: 440882540984883741.950751358174415911477197127174937537949264617286152397021983
submitted by: apkorte
submitted at: 2026-10-23 01:09:30 UTC
last updated: 2026-10-23 01:09:30 UTC

Witness: 20 independent points log in to add more points →

(818956602960454, 4308169133732844955246)
(-1431874966541096, 11512131136784976851146)
(18152204883662524/25, 95414444809471109206322/125)
(5112051738383968, -355244265269146022197526)
(606625636774444, -5659968069347610718070)
(618737802711406, 5129222683913128436356)
(635686880794252, -4383211603849135106966)
(738355474223836, 836451365083764881626)
(405531718438588, -14036764933267340332166)
(735970001282908, 790587998955335642698)
(670422231351250, 2850416403949994063086)
(630388937117716, -4616750936069890205462)
(20380927323037750, -2904146210045188147560164)
(1466336522163436, -39837187613069425545974)
(732853314777052, -751921025646950041526)
(1083989836648600, -17867437693583833912886)
(89298993607323724/121, -1103384783165386133681794/1331)
(723883440237244, 794635862846888963530)
(-302461004119052, 35138457448505744709538)
(-152431663412420, -31921002730193874453254)

Commentary

Specialization at T = -5/69 of the rank-17 fibration of X1008 given in curve#3589. Found 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 2026-10-23 01:09:30 UTC · history