curve #5290

y² + xy = x³ − 20940581812151627500486215467445x + 37494540228996028974655702164236054026654985121

a-invariants: [1, 0, 0, -20940581812151627500486215467445, 37494540228996028974655702164236054026654985121]
c-invariants: [1005147926983278120023338342437361, -32395282757852570541824417144817130714037420800585]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 135712212045423145251384547210301605950186052761297293697205002
discriminant (Δ): -19635951463375909578337760225107225475718071959718082113330054547071643894216794705143756750848
Faltings height: 16.8869
naive height: 228.0042
primes of bad reduction: 2, 3, 19, 23, 31, 43, 79, 109, 487, 95829791, 227655920844445350211, 424418250966331000111
regulator: 36321952605783929195.5788288753456768487971906706749586637555818413705312581175
submitted by: hdeping
submitted at: 2026-10-21 12:16:38 UTC
last updated: 2026-10-21 12:16:38 UTC

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

(1655263610727652863/4, -2129483655943492815982738923/8)
(15960447651381906, 1941384113065954164730071)
(-3613088176909170, -256881270745182106642029)
(25029173651998644246/9025, -23480860259524900603657009179/857375)
(3911985355314072, -124269326501700362747361)
(3107858483505506791986/1343281, 58880753902397213969773385665617/1556862679)
(2651753298501344754102/395641, 110799567239185011900554833605711/248858189)
(5053067890245522, 246379466796375201251799)
(2775701525778210, -27480546789250337457369)
(10692436586281395/4, -199007837424548000618733/8)
(116283014705616317634/38809, 310597684395493437373143614523/7645373)
(1879658838622171042/729, 499078738455176465483525317/19683)
(50360707513083954/25, 7379904241505740181210907/125)
(2705931110919666, 25372561970754823098711)
(310996448671051282/81, 85306355407698246593030591/729)
(2685711064987218, -25026783584153277390441)
(91123424207306577666/38809, -272548125617074628356793105157/7645373)
(4092385239179474, 142601935885104975539735)
(3937815491011872, 126868823369214032477169)
(5070508512905010, 248350964120058770358231)

Commentary

fiber of the x1092 K3 fibration family (file 167_x1092_t0_0085.fiber), E(u): y^2 = x^3 + A(u)*x + B(u), specialized at T = u = -9/19. A(u) = (-1383436870888675706200107)+(-5515600836340711948476888)*u^1+(716527474181252185372992)*u^2+(19599652286308507036273344)*u^3+(-10348738112891504390620680)*u^4+(11954627818857380106162864)*u^5+(-12819504668705779187131920)*u^6+(-9282889399122096275139840)*u^7+(-1414273781478875646923712)*u^8 ; B(u) = (623151887106655251853879329773846694)+(3757259803598750736604123706461347144)*u^1+(3844429892270677596667121714023090128)*u^2+(-16349006985383236996062158267218060800)*u^3+(-30783616351291087563896953575812615016)*u^4+(27168228250111848663166931761100037840)*u^5+(103348463813083983014377415708098752432)*u^6+(-94509325719259964077509140015970425664)*u^7+(-43457863230894151560075098944198688496)*u^8+(25171285331836505034229860335000995200)*u^9+(23456140237510943194372729657306631040)*u^10+(6556407886828091399912763281517445632)*u^11+(637439003051955708260756861597134080)*u^12. Rank exactly 20: 20 independent witness points (17 from mod-p Shioda fake 2-descent section lift, 3 more from kappa(z*)-covers of the 20 shallowest 2*Lambda_17 coset classes, ratpoints H=1e6, exact halving), canonical height Gram det != 0; Bober analytic rank bound 20 (kappa=2.8, GRH), rank is exactly 20.

last edited by hdeping at 2026-10-21 12:16:38 UTC · history