MAGMA online: all RANK =0,

but indeed MAGMA sometime rank is defferent from Cremona's tables ,I meet one times before

Qx := PolynomialRing(Rationals());

E00:=EllipticCurve(x^3 - 9122*x + 106889
);
E00;
MordellWeilShaInformation(E00);
DescentInformation(E00);
Generators(E00) ;
Q00, reps := IntegralPoints(E00);
Q00;
TorsionSubgroup(E00);

Elliptic Curve defined by y^2 = x^3 - 9122*x + 106889 over Rational Field

Torsion Subgroup = Z/4
Analytic rank = 0
==> Rank(E) = 0

[ 0, 0 ]
[]
[]

Torsion Subgroup = Z/4
Analytic rank = 0
==> Rank(E) = 0

[ 0, 0 ]
[]
[]
Warning: rank computed (0) is only a lower bound
(It may still be correct, though)
[ (-32 : 605 : 1) ]
[ (-32 : 605 : 1), (89 : 0 : 1) ]
Abelian Group isomorphic to Z/4
Defined on 1 generator
Relations:
4*$.1 = 0

=========================

Qx := PolynomialRing(Rationals());

E00:=EllipticCurve(x^3 - x^2 - 42144*x + 66420);
E00;
MordellWeilShaInformation(E00);
DescentInformation(E00);
Generators(E00) ;
Q00, reps := IntegralPoints(E00);
Q00;
TorsionSubgroup(E00);

Elliptic Curve defined by y^2 = x^3 - x^2 - 42144*x + 66420 over Rational Field

Torsion Subgroup = Z/4
Analytic rank = 0
==> Rank(E) = 0

[ 0, 0 ]
[]
[]

Torsion Subgroup = Z/4
Analytic rank = 0
==> Rank(E) = 0

[ 0, 0 ]
[]
[]
Warning: rank computed (0) is only a lower bound
(It may still be correct, though)
[ (-84 : 1734 : 1) ]
[ (-84 : 1734 : 1), (205 : 0 : 1) ]
Abelian Group isomorphic to Z/4
Defined on 1 generator
Relations:
4*$.1 = 0

===============

Qx := PolynomialRing(Rationals());

E00:=EllipticCurve(x^3 - x^2 - 168615*x + 21827700
);
E00;
MordellWeilShaInformation(E00);
DescentInformation(E00);
Generators(E00) ;
Q00, reps := IntegralPoints(E00);
Q00;
TorsionSubgroup(E00);

Elliptic Curve defined by y^2 = x^3 - x^2 - 168615*x + 21827700 over Rational
Field

Torsion Subgroup = Z/4
Analytic rank = 0
==> Rank(E) = 0

[ 0, 0 ]
[]
[]

Torsion Subgroup = Z/4
Analytic rank = 0
==> Rank(E) = 0

[ 0, 0 ]
[]
[]
Warning: rank computed (0) is only a lower bound
(It may still be correct, though)
[ (-45 : -5415 : 1) ]
[ (-45 : -5415 : 1), (316 : 0 : 1) ]
Abelian Group isomorphic to Z/4
Defined on 1 generator
Relations:

# 4*$.1 = 0

Qx := PolynomialRing(Rationals());

E00:=EllipticCurve(x^3 - 210386*x + 32627329);
E00;
MordellWeilShaInformation(E00);
DescentInformation(E00);
Generators(E00) ;
Q00, reps := IntegralPoints(E00);
Q00;
TorsionSubgroup(E00);

Elliptic Curve defined by y^2 = x^3 - 210386*x + 32627329 over Rational Field

Torsion Subgroup = Z/4
Analytic rank = 0
==> Rank(E) = 0

[ 0, 0 ]
[]
[]

Torsion Subgroup = Z/4
Analytic rank = 0
==> Rank(E) = 0

[ 0, 0 ]
[]
[]
Warning: rank computed (0) is only a lower bound
(It may still be correct, though)
[ (-24 : -6137 : 1) ]
[ (-24 : -6137 : 1), (337 : 0 : 1) ]
Abelian Group isomorphic to Z/4
Defined on 1 generator
Relations:
4*$.1 = 0