Commentary history

← curve #5205

1 edit.

hdeping · 2026-10-21 09:53:05 UTC

fiber 66 of the x1152 K3 fibration, E(u): y^2 = x^3 + A(u)*x + B(u), specialized at T = u = -40/7. A(u) = (-5523027425962212672208603351992987)+(11931822446710335547122692437257792)*u^1+(1691960323687093239746136961856976)*u^2+(-8325704248879965860362076192156544)*u^3+(-5170257801929882674508264302171680)*u^4+(-1860050058415221674720962229101056)*u^5+(-451161875487198592888382302550784)*u^6+(-28773514612260767178393000394752)*u^7+(-699834632413116089430954830592)*u^8 ; B(u) = (159000839890638574975169504773663511028739656150966)+(-501191565495384999221133904426408580841668740373184)*u^1+(197914167405983542084543638913806542299272417123984)*u^2+(428503287351301669607375291308003425413511660240000)*u^3+(-123040423514928090425285111998045420124005272593760)*u^4+(-105923509944757362498894072270789345100066091192832)*u^5+(121741900375944705432887583169914040801120454358528)*u^6+(107629873378225312023391466690280996994598937679872)*u^7+(30533984382625449198465286646323270119548502612480)*u^8+(3821382734656637004860508003894628059868846448640)*u^9+(326817339610894946622160643396035371237311287296)*u^10+(15018242786000481569679654452206674531395567616)*u^11+(280196251925076778185717521863443625575604224)*u^12. Rank 21 (exactly): 21 independent witness points (17 from mod-p Shioda fake 2-descent section lift, 4 more from kappa(z*)-covers of the 20 shallowest 2*Lambda_17 coset classes, ratpoints H=1e8, exact halving), 21x21 canonical height det = 4.79e20 != 0; Bober analytic rank bound 21 (kappa=2.8, GRH), root number -1, so rank is exactly 21.