curve #5291

y² + xy + y = x³ − 143592781862499793324405776968x + 25342092102508355199473656337230793829833402

a-invariants: [1, 0, 1, -143592781862499793324405776968, 25342092102508355199473656337230793829833402]
c-invariants: [6892453529399990079571477294441, -21895567576567229231025533175352525226192001205]
rank (lower bound): ≥ 21
torsion subgroup: trivial
conductor (N): 347225358872056718656732779074095147946415002428488763635566127923802
discriminant (Δ): -87953451086090190861214750081495749948550260907471014585442400070033798566310911116298868
Faltings height: 15.7832
naive height: 213.4052
primes of bad reduction: 2, 3, 11, 13, 19, 23, 31, 83, 103, 63044070442806421217, 55427010356864589386966709789244988519
regulator: 94517495689343595045.198328595843957787821887510488327201515183774733886362252390
submitted by: hdeping
submitted at: 2026-10-21 12:16:43 UTC
last updated: 2026-10-21 12:16:43 UTC

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

(-777510123212470080/2209, 589861960861152340390759178/103823)
(-96672971205236590722/237169, 464143414155728433068612978158/115501303)
(-71766784482216119/225, 20993567570667944973640159/3375)
(-912595527070687386/2209, 390607477928453572876734665/103823)
(221756919323122647/1156, -86609392685202199731554291/39304)
(7461276778937115, -643683228197783380029419)
(-90896127449503689/289, -30835392399086723657472631/4913)
(-413457410370519, -3745967682384304947995)
(-393668820672279, -4567407432509326518395)
(1911611948400578493/8464, 1639297937809599462234078503/778688)
(1096732930583305461/13456, -5877761213316988023793220395/1560896)
(23984871932403827/121, 2875505999087342634284031/1331)
(61529269968172624937244/420373009, 23541470643849722580028964353013410/8618907803527)
(2697900982215201/25, -416512764346398225370339/125)
(-11034014518596021/25, 206972788614210096475523/125)
(1134303419197006621/144, 1206707582381323710534133621/1728)
(7191499850319927/49, -934886562034719181743815/343)
(357222594910887, 4430796649158270633961)
(410251022812287, 5956565483283074883841)
(565506812285049807/1369, 306074931922432127758397413/50653)
(72607369099181643/289, 11121559733425643815250453/4913)

Commentary

fiber of the x1092 K3 fibration family (file 222_x1092_t3_0357.fiber), E(u): y^2 = x^3 + A(u)*x + B(u), specialized at T = u = -9/34. A(u) = (-176037064306992808566507)+(5915756000685239016284448)*u^1+(-74739832214501898117378864)*u^2+(407193784801308814869840384)*u^3+(-357852151289462141058677280)*u^4+(-6136035049085750455308401664)*u^5+(20390581978882532274482816256)*u^6+(24085910297771228781276198912)*u^7+(-138434852508586604980011424512)*u^8 ; B(u) = (29353282297132394631805644617637606)+(-1465800220984727623268853753715246176)*u^1+(29684854589829915583090967820635629584)*u^2+(-302504492712046802509595254047692878080)*u^3+(1489300822836997474656472031816649171360)*u^4+(-1859111704531363363401367591512810029568)*u^5+(-8901302556884468646838941329449484892672)*u^6+(7835621959513555079061283351216613480448)*u^7+(5887180595638993725830206009454293117440)*u^8+(905676905416813548887689587645787310653440)*u^9+(-2692953602420758530115438575793251800838144)*u^10+(-4969979756041599501329003211795628338610176)*u^11+(18717651081077897840135894513712398618025984)*u^12. Rank exactly 21: 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=1e6, exact halving), canonical height Gram det != 0; Bober analytic rank bound 21 (kappa=2.8, GRH), rank is exactly 21.

last edited by hdeping at 2026-10-21 12:16:43 UTC · history