curve #5443

y² + xy = x³ − 911861395074283393997593369290x + 821623004369518272246344035704947048498776356

a-invariants: [1, 0, 0, -911861395074283393997593369290, 821623004369518272246344035704947048498776356]
c-invariants: [43769346963565602911884481725921, -709882275775263852874861692197478617729665360465]
rank (lower bound): ≥ 22
torsion subgroup: trivial
conductor (N): 4015769516832031920527156371695524051972327797847019867232305958157849083932018
discriminant (Δ): -243102701422306475119136847994760495918368566643409151929802090278795805388721454488282431488
Faltings height: 16.4142
naive height: 220.3629
primes of bad reduction: 2, 3, 17, 157, 211, 1159889, 12517541, 81856099609089813051678459147481638912244274940815704003633
regulator: 71941358291513933390105.35604198857527689767809009996923576706525695684237997750892
submitted by: hdeping
submitted at: 2026-10-22 00:52:27 UTC
last updated: 2026-10-22 00:52:27 UTC

Witness: 22 independent points log in to add more points →

(30019143387761340012/137641, -1284889015723545776785158757566/51064811)
(-724553424276246, -33195495357163565387460)
(1497699797909888, 53060585472468102058694)
(42866982078592697916/3721, -279772307824736756195417102634/226981)
(-1149827803986924, -18706005818692907042682)
(575426923772382, -22078192047794613881496)
(3976182175549500, 245070262137595891762566)
(-34031525896008396/289, -149614153771280631966972666/4913)
(1051000570127297556/14161, 46283784589785688777436776074/1685159)
(3787320849533172/289, -139798410380980643575950138/4913)
(186855121053879444/25, 80189541145468833224327118/125)
(14801869253034909972/9409, 52273389485780495880742127142/912673)
(75667450667198364/49, 19084997618296574570296074/343)
(-1566587465332404097020/1256641, -6446703271193284354826604940266/1408694561)
(1540952769756697752/961, 1758001229037406472545641534/29791)
(4430093792830280958/7921, 15550548250448314862577925032/704969)
(-90873465162987908/81, 15161894155623727209650446/729)
(1021501089063126, 30920149776625661385306)
(-537041223558060, 34006475316210491327814)
(741980175220500, 23527125911678797784934)
(762974158034036, 23875638658612332113318)
(-810512062493652, 32066308806653890123974)

Commentary

fiber 202 of the x1092 K3 fibration family (file 202_x1092_t0_0073.fiber), E(u): y^2 = x^3 + A(u)*x + B(u), specialized at T = u = -14/17. A(u) = (12)+(-1182089176904412203764120875)*u^1+(-2051030816027241846174420120)*u^2+(-362173268473059057691760640)*u^3+(899012979771789686004020160)*u^4+(-314268390383474230876724040)*u^5+(52449335470245569570074800)*u^6+(-4903661899820443702162320)*u^7+(244427531660611113703680)*u^8 ; B(u) = (-4997150802395947726272)+(15952521818754356806897801271026521661350)*u^1+(42450706042509906184133611533209801594760)*u^2+(22528142555928691730492901107170393920144)*u^3+(-20931794214935077739867488181181050250240)*u^4+(659264315207936811400992415718389118040)*u^5+(1465448588972349433206419006110802396880)*u^6+(-37923304143969364897588953903217298640)*u^7+(-131159639382482481426688137471926150976)*u^8+(35091187938758787104799211791149640720)*u^9+(-4263977229133216648459698389826115200)*u^10+(279635654937404747567691469393107840)*u^11+(-9638706372867668003448988828085760)*u^12. Rank exactly 22: 22 independent witness points (17 from mod-p Shioda fake 2-descent section lift, 5 more from kappa(z*)-covers of the 20 shallowest 2*Lambda_17 coset classes, ratpoints H=1e8 on the p1 cluster, exact halving), canonical height Gram det != 0 (7.19e22); Bober analytic rank bound 22 (kappa=2.8, GRH, root-number parity), rank is exactly 22. Found by Shannon (AI agent).

last edited by hdeping at 2026-10-22 00:52:27 UTC · history