2102.sagews

E = EllipticCurve([0,1,1,0,0]); E   # probeer ook [0,1,1,0,0]

Elliptic Curve defined by y^2 + y = x^3 + x^2 over Rational Field

cremona = E.mwrank(); print(cremona)

Curve [0,1,1,0,0] : Basic pair: I=16, J=-560 disc=-297216 2-adic index bound = 2 By Lemma 5.1(a), 2-adic index = 1 2-adic index = 1 One (I,J) pair Looking for quartics with I = 16, J = -560 Looking for Type 3 quartics: Trying positive a from 1 up to 2 (square a first...) (1,0,-8,12,-4) --nontrivial...(x:y:z) = (1 : 1 : 0) Point = [1:1:1] height = 0.565348563787389 Rank of B=im(eps) increases to 1 (1,0,4,4,0) --trivial Trying positive a from 1 up to 2 (...then non-square a) Trying negative a from -1 down to -1 Finished looking for Type 3 quartics. Mordell rank contribution from B=im(eps) = 1 Selmer rank contribution from B=im(eps) = 1 Sha rank contribution from B=im(eps) = 0 Mordell rank contribution from A=ker(eps) = 0 Selmer rank contribution from A=ker(eps) = 0 Sha rank contribution from A=ker(eps) = 0 Used full 2-descent via multiplication-by-2 map Rank = 1 Rank of S^2(E) = 1 Searching for points (bound = 8)...done: found points of rank 1 and regulator 0.0628165070874876 Processing points found during 2-descent...done: now regulator = 0.0628165070874876 Saturating (bound = 100)...done: points were already saturated. Transferring points from minimal curve [0,1,1,0,0] back to original curve [0,1,1,0,0] Generator 1 is [0:-1:1]; height 0.0628165070874876 Regulator = 0.0628165070874876 The rank and full Mordell-Weil basis have been determined unconditionally. (0.26 seconds)

gens = E.gens(); gens

[(0 : 0 : 1)]

E.integral_points(both_signs=True, verbose=True)

Using mw_basis [(0 : 0 : 1)] e1,e2,e3: 0.423159354826277 - 0.451506572928502*I 0.423159354826277 + 0.451506572928502*I -0.846318709652555 Minimal eigenvalue of height pairing matrix: 0.0628165070874876 x-coords of points on non-compact component with -1 <=x<= 1 [-1, 0, 1, 2] starting search of remaining points using coefficient bound 17 x-coords of extra integral points: [-1, 0, 1, 2, 21] Total number of integral points: 10 [(-1 : -1 : 1), (-1 : 0 : 1), (0 : -1 : 1), (0 : 0 : 1), (1 : -2 : 1), (1 : 1 : 1), (2 : -4 : 1), (2 : 3 : 1), (21 : -99 : 1), (21 : 98 : 1)]

E.torsion_points()

[(0 : 1 : 0)]

P = gens[0]; P

(0 : 0 : 1)

for n in range(-15,15): print(n*P)

(528157/1036324 : -1372862351/1054977832 : 1) (194438/38809 : 90160188/7645373 : 1) (-17139/36481 : 737371/6967871 : 1) (-3629/7569 : -729620/658503 : 1) (1740/361 : -83259/6859 : 1) (209/400 : 2527/8000 : 1) (-140/121 : -400/1331 : 1) (11/49 : -363/343 : 1) (21 : 98 : 1) (-2/9 : 1/27 : 1) (-3/4 : -9/8 : 1) (2 : -4 : 1) (1 : 1 : 1) (-1 : 0 : 1) (0 : -1 : 1) (0 : 1 : 0) (0 : 0 : 1) (-1 : -1 : 1) (1 : -2 : 1) (2 : 3 : 1) (-3/4 : 1/8 : 1) (-2/9 : -28/27 : 1) (21 : -99 : 1) (11/49 : 20/343 : 1) (-140/121 : -931/1331 : 1) (209/400 : -10527/8000 : 1) (1740/361 : 76400/6859 : 1) (-3629/7569 : 71117/658503 : 1) (-17139/36481 : -7705242/6967871 : 1) (194438/38809 : -97805561/7645373 : 1)