贴张Silverman书上的

上传的附件：

• 未命名.JPG （43.92kb，58次下载）

Y^2=x^3-4x^2+16MOD17

17+2,19/29都4MOD5，图中的所有点64

看来楼主在安全方面学术还不错,发表了这么主题

没人啊。。。刷墙涂鸦

密码白痴飘过，友情帮顶

多谢！

先贴点慢慢理解

Qx<x> := PolynomialRing(Rationals());
E1:=EllipticCurve( x^3-39);-

E1;
jInvariant(E1);
Genus(E1);

Q1, reps := IntegralPoints(E1);
Q1;
TorsionSubgroup(E1);-------------------------------------RANK=2，T=1
Mordell's curve   E: y^2 = x^3 - 39                       %%%
%%%                                                                       %%%
%%%    E_-00039: r = 2   t = 1   #III =  1                                %%%
%%%              E(Q) = <(4, 5)> x <(10, 31)>                             %%%
%%%              R =   2.7347941297                                       %%%
%%%               6 integral points                                       %%%
%%%                1. (4, 5) = 1 * (4, 5)                                 %%%
%%%                2. (4, -5) = -(4, 5)                                   %%%
%%%                3. (22, 103) = 1 * (4, 5) - 1 * (10, 31)               %%%
%%%                4. (22, -103) = -(22, 103)                             %%%
%%%                5. (10, 31) = 1 * (10, 31)                             %%%
%%%                6. (10, -31) = -(10, 31)   

Elliptic Curve defined by y^2 = x^3 - 39 over Rational Field
0
1
[ (4 : 5 : 1), (10 : 31 : 1), (22 : -103 : 1) ]
Abelian Group of order 1
==========================

E2:=EllipticCurve(x^3 +1331);---------------------------------RANK=1，T=2
E2;
jInvariant(E2);
Genus(E2);
TorsionSubgroup(E2);
Q2, reps := IntegralPoints(E2);
Q2;
E_+01331: r = 1   t = 2   #III =  1
          E(Q) = <(37, 228)> x <(-11, 0)>
          R =   3.2472475292
           3 integral points
            1. (-11, 0) = (-11, 0)
            2. (37, 228) = 1 * (37, 228)
            3. (37, -228) = -(37, 228)
Elliptic Curve defined by y^2 = x^3 + 1331 over Rational Field
0
1
Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0
[ (-11 : 0 : 1), (37 : -228 : 1) ]
=========================

E3:=EllipticCurve(x^3 - 9998);------RANK=3，T=1 无整点
E3;
jInvariant(E3);
Genus(E3);
TorsionSubgroup(E3);
Q3, reps := IntegralPoints(E3);
Q3;

r = 3   t = 1   #III =  1
          E(Q) = <(-71/4, 531/8)> x <(19/9, 2701/27)> x <(283/9, 5473/27)>
          R = 114.5499715889
           0 integral points
Elliptic Curve defined by y^2 = x^3 - 9998 over Rational Field
0
1
Abelian Group of order 1
[]

==========================
E6:=EllipticCurve(x^3 +1521);----------RANK=1，T=3，有三对整点
E6;
jInvariant(E6);
Genus(E6);
TorsionSubgroup(E6);
Q6, reps := IntegralPoints(E6);
Q6;
E_+01521: r = 1   t = 3   #III =  1
          E(Q) = <(12, 57)> x <(0, 39)>
          R =   1.9164245639
           6 integral points
            1. (0, 39) = (0, 39)
            2. (0, -39) = -(0, 39)
            3. (12, 57) = 1 * (12, 57)
            4. (12, -57) = -(12, 57)
            5. (52, 377) = (0, 39) - 1 * (12, 57)
            6. (52, -377) = -(52, 377)
Elliptic Curve defined by y^2 = x^3 + 1521 over Rational Field
0
1
Abelian Group isomorphic to Z/3
Defined on 1 generator
Relations:
    3*$.1 = 0
[ (0 : 39 : 1), (12 : -57 : 1), (52 : 377 : 1) ]

E5:=EllipticCurve(x^3 -4*x^2+16);-----RANK=？，T=1，整点？
E5;
jInvariant(E5);
Genus(E5);
TorsionSubgroup(E5);
Q5, reps := IntegralPoints(E5);
Q5;

E7:=EllipticCurve(x^3 + 7388);
E7;
jInvariant(E7);
Genus(E7);
TorsionSubgroup(E7);
Q7, reps := IntegralPoints(E7);
Q7;

E8:=EllipticCurve(x^3 +7428);
E8;

jInvariant(E8);
Genus(E8);
TorsionSubgroup(E8);
Q8, reps := IntegralPoints(E8);
Q8;

E9:=EllipticCurve(x^3 - 7416);
E9;
jInvariant(E9);
Genus(E9);
TorsionSubgroup(E9);
Q9, reps := IntegralPoints(E9);
Q9;

上传的附件：

• a.JPG （81.98kb，38次下载）
• b.JPG （26.92kb，38次下载）
• c.JPG （40.70kb，38次下载）

Qx<x> := PolynomialRing(Rationals());
E1:=EllipticCurve( x^3-39);
E1;
MordellWeilRank(E1);
RankBounds(E1);
MordellWeilShaInformation(E1);
F:=FaltingsHeight(E1) ;
F;
HasComplexMultiplication(E1);
DescentInformation(E1);
Generators(E1) ;
Q1, reps := IntegralPoints(E1);
Q1;
TorsionSubgroup(E1);

Elliptic Curve defined by y^2 = x^3 - 39 over Rational Field
2
2 2
Torsion Subgroup is trivial
Analytic rank = 2
The 2-Selmer group has rank 2
Found a point of infinite order.
Found 2 independent points.
After 2-descent:
    2 <= Rank(E) <= 2
    Sha(E)[2] is trivial
(Searched up to height 100 on the 2-coverings.)
[ 2, 2 ]
[ (22 : -103 : 1), (4 : 5 : 1) ]
[
    <2, [ 0, 0 ]>
]
-0.204821688386087651093858399724
true -3
Torsion Subgroup is trivial
Analytic rank = 2
The 2-Selmer group has rank 2
Found a point of infinite order.
Found 2 independent points.
After 2-descent:
    2 <= Rank(E) <= 2
    Sha(E)[2] is trivial
(Searched up to height 100 on the 2-coverings.)
[ 2, 2 ]
[ (22 : -103 : 1), (4 : 5 : 1) ]
[
    <2, [ 0, 0 ]>
]
[ (4 : 5 : 1), (10 : 31 : 1) ]
[ (4 : 5 : 1), (10 : 31 : 1), (22 : -103 : 1) ]
Abelian Group of order 1

火星
语言不通
路过

E00:=EllipticCurve( x^3-4*x^2+16);
E00;
MordellWeilShaInformation(E00);
DescentInformation(E00);
Generators(E00) ;
Q00, reps := IntegralPoints(E00);
Q00;
TorsionSubgroup(E00);

Elliptic Curve defined by y^2 = x^3 - 4*x^2 + 16 over Rational Field


mod17
0
0 0
Torsion Subgroup = Z/5
Analytic rank = 0
     ==> Rank(E) = 0
[ 0, 0 ]
[]
[]
Torsion Subgroup = Z/5
Analytic rank = 0
     ==> Rank(E) = 0
[ 0, 0 ]
[]
[]
[ (0 : 4 : 1) ]
[ (0 : 4 : 1), (4 : -4 : 1) ]
Abelian Group isomorphic to Z/5
Defined on 1 generator
Relations:
    5*$.1 = 0

====================
Qx<x> := PolynomialRing(Rationals());
E0:=EllipticCurve( x^3-2*x+16);
E0;
MordellWeilShaInformation(E0);

DescentInformation(E0);
Generators(E0) ;
Q0, reps := IntegralPoints(E0);
Q0;
TorsionSubgroup(E0);

Elliptic Curve defined by y^2 = x^3 - 2*x + 16 over Rational Field



mod17
1
1 1
Torsion Subgroup is trivial
Analytic rank = 1
     ==> Rank(E) = 1
The 2-Selmer group has rank 1
Found a point of infinite order.
After 2-descent:
    1 <= Rank(E) <= 1
    Sha(E)[2] is trivial
(Searched up to height 100 on the 2-coverings.)
[ 1, 1 ]
[ (0 : 4 : 1) ]
[
    <2, [ 0, 0 ]>
]
Torsion Subgroup is trivial
Analytic rank = 1
     ==> Rank(E) = 1
The 2-Selmer group has rank 1
Found a point of infinite order.
After 2-descent:
    1 <= Rank(E) <= 1
    Sha(E)[2] is trivial
(Searched up to height 100 on the 2-coverings.)
[ 1, 1 ]
[ (0 : 4 : 1) ]
[
    <2, [ 0, 0 ]>
]
[ (0 : 4 : 1) ]
[ (0 : 4 : 1) ]
Abelian Group of order 1

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

Qx<x> := PolynomialRing(Rationals());
E0:=EllipticCurve( x^3-x+16);

mod17



mod21
E0;
MordellWeilShaInformation(E0);

DescentInformation(E0);
Generators(E0) ;
Q0, reps := IntegralPoints(E0);
Q0;
TorsionSubgroup(E0);

Elliptic Curve defined by y^2 = x^3 - x + 16 over Rational Field
2
2 2
Torsion Subgroup is trivial
Analytic rank = 2
The 2-Selmer group has rank 2
Found a point of infinite order.
Found 2 independent points.
After 2-descent:
    2 <= Rank(E) <= 2
    Sha(E)[2] is trivial
(Searched up to height 100 on the 2-coverings.)
[ 2, 2 ]
[ (0 : 4 : 1), (1 : -4 : 1) ]
[
    <2, [ 0, 0 ]>
]

Torsion Subgroup is trivial
Analytic rank = 2
The 2-Selmer group has rank 2
Found a point of infinite order.
Found 2 independent points.
After 2-descent:
    2 <= Rank(E) <= 2
    Sha(E)[2] is trivial
(Searched up to height 100 on the 2-coverings.)
[ 2, 2 ]
[ (0 : 4 : 1), (1 : -4 : 1) ]
[
    <2, [ 0, 0 ]>
]
[ (1 : 4 : 1), (-1 : 4 : 1) ]
[ (-1 : -4 : 1), (0 : 4 : 1), (1 : -4 : 1), (16 : -64 : 1), (63 : 500 : 1), (65
: -524 : 1), (262016 : 134119436 : 1) ]
Abelian Group of order 1

上传的附件：

• 1.JPG （36.09kb，13次下载）
• 2.JPG （23.37kb，13次下载）
• 3.JPG （25.35kb，13次下载）
• 4.JPG （18.72kb，13次下载）