curve #4072

y² + xy + y = x³ − x² − 867676635677517843452605585515194512078234763210916155x + 311088453085744437630614178149759454244744099375091764759736060102972383223090347

a-invariants: [1, -1, 1, -867676635677517843452605585515194512078234763210916155, 311088453085744437630614178149759454244744099375091764759736060102972383223090347]
c-invariants: [41648478512520856485725068104729336579755268634123975425, -268780423466083194112850649733974015161115047674316478281129941320069430251192170625]
rank (lower bound): ≥ 20
torsion subgroup: ℤ/2ℤ
conductor (N): 6298072849453555625678659198919335015893455426961399018547798525692978105732842174850
discriminant (Δ): 207349507308655959429075239992235161493301950303762984422555754646003655731912246841381845272444622717259713761380871975356918479806824623048813922750464000000
Faltings height: 29.6088
naive height: 384.2066
primes of bad reduction: 2, 3, 5, 13, 17, 19, 23, 29, 41, 43, 67, 73, 83, 89, 109, 139, 173, 181, 919, 1217, 1931, 2459, 2719, 3533, 4259, 4993, 8081, 9629, 100547, 234673, 761667509
regulator: 6614713888757303096074.12946211246860621191527943551907107372093927786575789135
submitted by: wgxli
submitted at: 2026-10-09 04:23:58 UTC
last updated: 2026-10-09 04:23:58 UTC

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

(537103011622895827301431819, 2262879917470743071106446787409366490)
(537103128729420844962838859, 2204222833813179042510783877422460570)
(536356417084628017197580993, 50729759566961893004743480179178214246)
(536573605711769480061355309, 40512055549687972088843857197418919420)
(536709352064282539484141995, 33699449019549714467142878201479237178)
(536710871447946812379735019, 33620378495555528919122895565575351290)
(536855144983401242057788363, 25666165873392680713565493510944254490)
(536869994860979884698992519, 24779399027036577462368337784618683790)
(538614744165844130634110659, 17554267866789550587294636082525913570)
(539022616076533063605727819, 40671574362993856732208035581949486490)
(522520521472896609741116059, 610064503056977981593492440499497318170)
(526187767614964180780636879, 463794314862434606644991859132030248810)
(533995395734437454642213579, 149966261911684362906877564477398960410)
(534472258814357217051407809, 130481396927432013832312323134367335270)
(548944778074162872374777059, 448460772977082916517595712994736078170)
(551391844982372645913465019, 547659148376721415690757560730867153690)
(395576813832973677423270019, 5454901988140914932642746803559810118690)
(-492354403991238515008430621, 24878501303843316830319622042863486782810)
(9149673582550012756517405179, 870833521938189470864755667965125484684010)
(1435779919386465755736295915/4, 54192126177637093464498019511548432930969/8)

Commentary

specialization at T = 60273/42619 of the rank-9 fibration of X808 given by: y^2 + (367T^2 + 35464T + 23700)xy = x(x + (51T - 727)(635T - 429)(6T + 13)(55T + 59))(x + (13T - 25)(2548T - 55)(375 - 272T)).

last edited by wgxli at 2026-10-09 04:23:58 UTC · history