curve #4122

y² = x³ + x² − 7031654147618623172800005930266726136805275621730972807108x − 8707579298767806475347488524279490571097769743056213158717287231301799085776884343712

a-invariants: [0, 1, 0, -7031654147618623172800005930266726136805275621730972807108, -8707579298767806475347488524279490571097769743056213158717287231301799085776884343712]
c-invariants: [337519399085693912294400284652802854566653229843086694741200, 7523348514135384794700230082952363458914309584234166461214919040444835031052707904520000]
rank (lower bound): ≥ 21
torsion subgroup: ℤ/2ℤ
conductor (N): 23006300323005732981493739280611137630592891352244258300189393439988018703931591002792570449685679400
discriminant (Δ): 22218395841715975794642079717683511633155910881079796397951283871386835827367890055819736043906438709829251329885176831079278845354158613902162717789704164551681175180676000000
Faltings height: 32.3354
naive height: 411.2069
primes of bad reduction: 2, 3, 5, 7, 13, 23, 41, 53, 97, 257, 347, 353, 383, 419, 433, 563, 587, 701, 1429, 1889, 2053, 4099, 4241, 5557, 8221, 9901, 12619, 29863, 33347, 240677, 26559851, 48997456307
regulator: 11939510175354349117442157.284046983717954453262573788281236800696860726774814728
submitted by: wgxli
submitted at: 2026-10-09 21:22:10 UTC
last updated: 2026-10-09 21:22:10 UTC

Witness: 21 independent points log in to add more points →

(84476485629783532881197151103, 361330671470058637554620608604750557126950)
(84479764095637331817474081388, 421542000061412183849481655409367236445600)
(-2611475443014951828586587022, 3104448917419609320521192168980260278293050)
(-3232065524096570632478232022, 3739709172188638615458419502647576885158050)
(89840112949107979213846833638, 9202687687884738594607361620750866660879150)
(90182768243683439345764873112, 9518859065344408934987964865828019020807608)
(-11949880080766138854780968146, 8579825524652597400350990967773147164905894)
(-15708632027758156481371800766, 9893119256175598286092269399944326270853526)
(-77960804398404128054555813272, 8102399211864686037744830322280365767773200)
(90787475170738004539734915992, 10060311176593897981867931908491547419924552)
(-33252815748615835663650037762, 13723899868691227106953372181089715670854550)
(-34574787828680725505834983826, 13895291216237354744528289107584497267770214)
(-62468115192008565743658261286, 13666727351159029410059368481156644228458354)
(110756244634159579187180837228, 23898402779850930223400048097033531733660800)
(173061710980693886329527237983, 62909793700406138325485311319901130399684350)
(428903624404334142935275869503, 275455586694685874825991060981643257572527350)
(3219645791785090319435025421928, 5775169551736226110509726080555079173804550000)
(-28427070729319947062477015876542/361, 51655131888147656397017407804530384818842017150/6859)
(119229892866799409027069275590947/169, 1292655143976275380733335656840673477952686987200/2197)
(-80828458479821445792447851118802/6241, 4415065798680142358708754695487378496865596165050/493039)
(5442833953675304348828264963584088/19321, 383263730954747057207346217125065779811116189979400/2685619)

Commentary

specialization at T = 77/159861 of the rank-9 fibration of X1152 given by: y^2 + (686021T^2 + 247258T - 9723)xy = x(x + (179T - 128)(447511 - 870090T)(66T + 113))(x + (1495T - 1654)(2139T - 562)(98T - 29)(153T + 116)).

last edited by wgxli at 2026-10-09 21:22:10 UTC · history