curve #631

y² + xy = x³ − 428377857530303119736330313x + 2949746475279594219167674546858042856217

a-invariants: [1, 0, 0, -428377857530303119736330313, 2949746475279594219167674546858042856217]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 2368241775053952045429990091103904061860334234693819877055450 ★ record for rank ≥ 20
discriminant (Δ): 1272243771415190417871760744895557443313119481010856539704955831996832663060480000
Faltings height: 14.2875
naive height: 195.5797
primes of bad reduction: 2, 3, 5, 7, 13, 17, 31, 139, 4799, 14731, 33503099826889072146871895726559589117904669
regulator: 233136076300214429.348343986911297193193381616647635174217931682493643026721687
submitted by: wgxli
submitted at: 2026-09-07 14:08:52 UTC
last updated: 2026-09-07 14:08:52 UTC

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

(387188893249866/25, -686176108848292463529/125)
(74443170755481402/3721, -11089590018295428820126401/226981)
(877723521785651181641734248707226/44557976433273538225, -13807216612898018568066031551228808491260089744731/297432340911279942286530260375)
(84490415179616562017366178/1306638744889, 740224465514277249179551449817593044193/1493596536423952787)
(1046916208472176582986605082/34541938263121, -27091109445955584828503129377160328314859/203011226695607002919)
(-847064236921289/36, 332963921221188530279/216)
(1526626576568237184738961578/98094385956121, -7027443063376463685155927425780458493545/971552401144156931581)
(-1537782366795674774335985622/126444020241289, -113396989407156180268529507830203457405617/1421829247059890380837)
(171972520691803981338/2505889, -2160977100369985025083598580147/3966822287)
(-126659915982629740662/6056521, 783341151320391247647121470447/14905098181)
(26064331754963231230785444259722/1555509039806631361, -42343403211676698269334319437541499933797129581/1940031622646678780428884641)
(-11880434541469200882/619369, 31246683257504069355649203597/487443403)
(17820845995646660711/3407716, 183675930572509029012216929039/6290643736)
(-2901362773221366796550880582/315018730020289, -437183398988989135552137325925823215806083/5591194039766014733663)
(-98532257680719902201805436847718/4400020576226005201, -334422515332876239372205704236241506456894225301/9229582605556711793561147801)
(1465699571174526081661952838157242/86329285708007774041, 19162053189074183685812552663029801447669077713751/802116077444790140110050618589)
(2122799193555989876864276682/72765011518081, 76730321735344409268017328055125234571779/620703084617006787521)
(95225808916788017217126142458/5045788382566369, 14278125961229803989137801131251682383809043/358421083825953658998097)
(984351073633586022875952/35251189009, 747666339053950776484959134903366349/6618516490006777)
(53582524302750575019269755818/3232663525371289, 3690548207950672282631904793367815768794327/183797988685552624340413)

Commentary

Specialization at T = 3/4 of the rank-15 fibration of X948 given by: y^2 + (Lx+B)y = x(x+D)(x+E), where B = pE + qD + pq(L-p-q), L = 74655330T^2 - 93107608T + 34781578, p = -255200T(725T - 619), q = 841321T - 86021, D = 1000T(648939726360T^3 - 1001134894922T^2 + 373250670277T - 34456851), E = -52(210701445490T^3 + 2107937211893T^2 - 2422822069031T + 454446440448).

last edited by wgxli at 2026-09-17 17:36:14 UTC · history