[was@modular rabi]$ [was@modular rabi]$ Magma V2.7-3      Fri Nov 24 2000 17:51:25 on modular  [Seed = 355590042]
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Loading startup file "/home/was/modsym/init-magma.m"

C               IndexGamma0     R               ellap           idxG0
CS              MS              S               factormod       modcharpoly
DC              ND              Tn              factorpadic     padiccharpoly
ES              NS              Z               fcp             qexp
F               Q               charpoly        fn              x
> E:=EC("65A");
> E
> ;
Elliptic Curve defined by y^2 + x*y = x^3 - x over Rational Field
> W:=WeierstrassForm(E);

>> W:=WeierstrassForm(E);
      ^
User error: Identifier 'WeierstrassForm' has not been declared or assigned
>  W:=Weierstrass(E);
> W;
[ -1323, 3942 ]
> W:=EllipticCurve([ -1323, 3942 ]);
> W;
[ -1323, 3942 ]
> W:=EllipticCurve([ -1323, 3942 ]);
> W;
Elliptic Curve defined by y^2 = x^3 - 1323*x + 3942 over Rational Field
> EW;

>> EW;
   ^
User error: Identifier 'EW' has not been declared or assigned
> E;
Elliptic Curve defined by y^2 + x*y = x^3 - x over Rational Field
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> E;
Elliptic Curve defined by y^2 + x*y = x^3 - x over Rational Field
> G,f := MordellWeilGroup(E);
> f(G.2);
(-1 : 1 : 1)
> Type($1);
CrvEllPt
> K<x>:=FractionField(PolynomialRing(Rationals()));

>> K<x>:=FractionField(PolynomialRing(Rationals()));
         ^
User error: Identifier 'FractionField' has not been declared or assigned
>  K<x>:=FieldOfFractions(PolynomialRing(Rationals()));
> K;
Rational function field of rank 1 over Rational Field
Variables: x
> Type(K);
FldFunRat
> P:=f(G.2);
> P;
(-1 : 1 : 1)
> P[1];
-1
> 2*P;
(4 : -10 : 1)
> Attach("rabi-11-24-00.m");
> G;
Abelian Group isomorphic to Z/2 + Z
Defined on 2 generators
Relations:
2*G.1 = 0
>  h := RabiFunction(E, f(G.2));
> h;
(x^2 + 34/25*x + 9/25)/(x^2 - 8*x + 16)
> f(G.1);
(0 : 0 : 1)
> Evaluate(h,0);
9/400
> factor($1);
[ <3, 2>, <2, -4>, <5, -2> ]
> h := RabiFunction(E, f(G.2));
> RabiFunction;
Intrinsic 'RabiFunction'

Signatures:

(<CrvEll> E, <CrvEllPt> P) -> FldFunRat

The rational function (x - x(P))*(x-x(3*P))/(x-x(2*P))^2.


> Order(f(G.2));
0
> h := RabiFunction(E, f(G.2));
> h;
(x^2 + 34/25*x + 9/25)/(x^2 - 8*x + 16)
> P:=f(G.2);
> P;
(-1 : 1 : 1)
> 3*P;
(-9/25 : 96/125 : 1)
> Type(P);
CrvEllPt
> F<y>:=FunctionField(E);
> F;
Algebraic function field defined over Rational Field by
$.1^2 + x*$.1 - x^3 + x
> R<x>:=PolynomialRing(Q);
> F<y>:=FunctionField(E);
> F;
Algebraic function field defined over Rational Field by
$.1^2 + x*$.1 - x^3 + x
> G;
Abelian Group isomorphic to Z/2 + Z
Defined on 2 generators
Relations:
2*G.1 = 0
> f(G.1);
(0 : 0 : 1)
> f(G.2);
(-1 : 1 : 1)
> h;
(x^2 + 34/25*x + 9/25)/(x^2 - 8*x + 16)
> P;
(-1 : 1 : 1)
> 2*P;
(4 : -10 : 1)
> 3*P;
(-9/25 : 96/125 : 1)
> G;
Abelian Group isomorphic to Z/2 + Z
Defined on 2 generators
Relations:
2*G.1 = 0
> h:=RabiFunction(E,P);
> h;
(x^2 + 34/25*x + 9/25)/(x^2 - 8*x + 16)
> Evaluate(h,0);
9/400
> Evaluate(RabiFunction(E,2*P),0);
130662006784/45120132225
> factor($1);
[ <2, 20>, <353, 2>, <3, -2>, <5, -2>, <7, -4>, <17, -4> ]
> Evaluate(RabiFunction(E,3*P),0);
37988629075871938150281/119283737148688367507344
> factor(Evaluate(RabiFunction(E,3*P),0));
[ <3, 10>, <7, 8>, <173, 2>, <1931, 2>, <2, -4>, <353, -4>, <692917, -2> ]
> factor(Evaluate(RabiFunction(E,4*P),0));
[ <2, 4>, <17, 8>, <41, 2>, <191, 4>, <775152793, 2>, <3, -2>, <5, -2>, <7, -4>, <67, -2>, <353, -2>, <577, -2>, <149057, -4> ]
> factor(Evaluate(RabiFunction(E,5*P),0));
[ <3, 2>, <11, 8>, <23, 2>, <37, 4>, <61, 2>, <101, 2>, <643, 4>, <997, 2>, <141488069, 2>, <2, -4>, <5, -4>, <73, -4>, <3517, -2>, <966937, -4>, <226761978977, -2> ]
> Q:=P;factor(Q[1]*(3*Q[1]));
> Q:=P;factor(Q[1]*(3*Q[1]));
[ <3, 1> ]
> Q:=P;factor(Q[1]*((3*Q)[1]));
[ <3, 2>, <5, -2> ]
> Q:=7*P;factor(Q[1]*((3*Q)[1]));
[ <3, 2>, <29, 4>, <31, 2>, <131, 4>, <757, 2>, <4523, 2>, <24371, 2>, <32579, 2>, <808455370189231, 2>, <5, -2>, <13, -4>, <457, -2>, <733, -4>, <2857, -2>, <55544655533, -2>, <216284387469461, -2> ]
> Q:=6*P;factor(Q[1]*((3*Q)[1]));
[ <2, 4>, <353, 4>, <16210522753, 2>, <19729076361697, 2>, <3, -6>, <5, -4>, <7, -8>, <167, -2>, <173, -2>, <1931, -2>, <692917, -2>, <2312150887, -2> ]
> Q:=P;factor(Q[1]*((3*Q)[1]));
[ <3, 2>, <5, -2> ]
> P;
(-1 : 1 : 1)
> 3*P;
(-9/25 : 96/125 : 1)
> Z<a,b>:=FieldOfFractions(PolynomialRing(Rationals(),2));
> F<y>:=FunctionField(E);
> P:=[x,y,1];
> P:=E!P;

>> P:=E!P;
       ^
Runtime error in '!': Cannot coerce element into ring
> P;
[ x, y, 1 ]
> E![x,y];

>> E![x,y];
    ^
Runtime error in '!': Cannot coerce element into ring
> Type(x);
RngUPolElt
> Type(y);
FldFunElt
> E![Parent(y)|x,y];

>> E![Parent(y)|x,y];
    ^
Runtime error in '!': Cannot coerce element into ring
> E;
Elliptic Curve defined by y^2 + x*y = x^3 - x over Rational Field
> y^2 + x*y - (x^3 - x);
0
> F:=BaseExtend(E,Parent(y));
> generic:=E![Parent(y)|x,y];

>> generic:=E![Parent(y)|x,y];
             ^
Runtime error in '!': Cannot coerce element into ring
> generic:=F![Parent(y)|x,y];
> generic;
(x : y : 1)
> 3*generic;
((1/9*x^9 + 4/3*x^7 + 2/9*x^6 + 10/3*x^5 + 4/3*x^4 - 35/9*x^3 - 2/3*x^2 + x)/(x^8 + 2/3*x^7 - 35/9*x^6 - 4/3*x^5 + 10/3*x^4 - 2/9*x^3 + 4/3*x^2 + 1/9) : (1/27*x^12 + 1/27*x^11 - 22/27*x^10 - 55/9*x^8 - 22/9*x^7 + 79/27*x^6 + 67/27*x^5 - 175/27*x^4 - 55/27*x^3 + 89/27*x^2 + 4/27*x - 1/9)/(x^12 + x^11 - 17/3*x^10 - 107/27*x^9 + 31/3*x^8 + 10/3*x^7 - 37/9*x^6 + 4/3*x^5 - 11/3*x^4 + 1/9*x^3 - 2/3*x^2 - 1/27)*y + (-1/27*x^13 - 26/27*x^11 - 1/3*x^10 - 92/27*x^9 - 20/9*x^8 + 182/27*x^7 + 97/27*x^6 - 188/27*x^5 - 44/27*x^4 + 2*x^3 - 1/27*x^2 + 1/9*x)/(x^12 + x^11 - 17/3*x^10 - 107/27*x^9 + 31/3*x^8 + 10/3*x^7 - 37/9*x^6 + 4/3*x^5 - 11/3*x^4 + 1/9*x^3 - 2/3*x^2 - 1/27) : 1)
> x_coord:=$1[1]; x_coord:=$1[1]; 
> x_coord;
(1/9*x^9 + 4/3*x^7 + 2/9*x^6 + 10/3*x^5 + 4/3*x^4 - 35/9*x^3 - 2/3*x^2 + x)/(x^8 + 2/3*x^7 - 35/9*x^6 - 4/3*x^5 + 10/3*x^4 - 2/9*x^3 + 4/3*x^2 + 1/9)
> factor(x_coord);

>> factor(x_coord);
         ^
Runtime error in 'factor': Bad argument types
Argument types given: FldFunElt
> Factorization(Numerator(x_coord));

>> factor(Numerator(x_coord));
                   ^
Runtime error in 'Numerator': Bad argument types
Argument types given: FldFunElt
> Factorization(Numerator(x_coord));

>> Factorization(Numerator(x_coord));
                          ^
Runtime error in 'Numerator': Bad argument types
Argument types given: FldFunElt
>  Factorization(PolynomialRing(Rationals())!Numerator(x_coord));

>>  Factorization(PolynomialRing(Rationals())!Numerator(x_coord));
                                                       ^
Runtime error in 'Numerator': Bad argument types
Argument types given: FldFunElt
> x_coord;
(1/9*x^9 + 4/3*x^7 + 2/9*x^6 + 10/3*x^5 + 4/3*x^4 - 35/9*x^3 - 2/3*x^2 + x)/(x^8 + 2/3*x^7 - 35/9*x^6 - 4/3*x^5 + 10/3*x^4 - 2/9*x^3 + 4/3*x^2 + 1/9)
> factor(1/9*x^9 + 4/3*x^7 + 2/9*x^6 + 10/3*x^5 + 4/3*x^4 - 35/9*x^3 - 2/3*x^2 + x);
[
<x, 1>,
<x^4 + 6*x^2 + x - 3, 2>
]
1/9
> factor(x^8 + 2/3*x^7 - 35/9*x^6 - 4/3*x^5 + 10/3*x^4 - 2/9*x^3 + 4/3*x^2 + 1/9);
[
<x^4 + 1/3*x^3 - 2*x^2 - 1/3, 2>
]
1
> P;
[ x, y, 1 ]
> f(G.2);
(-1 : 1 : 1)
> f(G.1);
(0 : 0 : 1)
> factor(Evaluate(RabiFunction(E,2*P),0));

>> factor(Evaluate(RabiFunction(E,2*P),0));
                                   ^
Runtime error in '*': Bad argument types
Argument types given: RngIntElt, SeqEnum[FldFunElt]
> P:=f(G.2);
> P;
(-1 : 1 : 1)
> Evaluate(RabiFunction(E,2*P),0);
[ <2, 20>, <353, 2>, <3, -2>, <5, -2>, <7, -4>, <17, -4> ]
> Evaluate(RabiFunction(E,2*P),0);
130662006784/45120132225
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> ////////////////////////////////
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> E:=EC("82A");
> G,f:=MordellWeilGroup(E);
> G;
Abelian Group isomorphic to Z/2 + Z
Defined on 2 generators
Relations:
2*G.1 = 0
> Evaluate(RabiFunction(E,f(G.2)),f(G.1)[1]);
9/4
> T:=f(G.1);
> P:=f(G.2);
> T;
(1 : -1 : 1)
> P;
(0 : 0 : 1)
> E;
Elliptic Curve defined by y^2 + x*y + y = x^3 - 2*x over Rational Field
> Evaluate(RabiFunction(E,2*f(G.2)),f(G.1)[1]);
1500625/576
> factor($1);
[ <5, 4>, <7, 4>, <2, -6>, <3, -2> ]
> Evaluate(RabiFunction(E,2*f(G.2)),2*f(G.1)[1]);
0
> f(2*G.1);
(0 : 1 : 0)
> Evaluate(RabiFunction(E,2*f(G.2)),f(G.1)[1]/f(G.1)[3]);
1500625/576
> factor($1);
[ <5, 4>, <7, 4>, <2, -6>, <3, -2> ]
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>   // RANK 2
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> E:=EC("1088J1");
> G,f:=MordellWeilGroup(E);
> G;
Abelian Group isomorphic to Z/2 + Z + Z
Defined on 3 generators
Relations:
2*G.1 = 0
> P1:=f(G.2);
> P2:=f(G.3);
> T:=f(G.1);
> T;
(3 : 0 : 1)
> P1;
(-1 : 8 : 1)
> P2;
(-6 : 3 : 1)
> Evaluate(RabiFunction(E,P1),T[1]/T[3]);
20736/289
> factor(20736/289);
[ <2, 8>, <3, 4>, <17, -2> ]
> Evaluate(RabiFunction(E,P2),T[1]/T[3]);
3017043937296/907505804276569
> factor(3017043937296/907505804276569);
[ <2, 4>, <3, 10>, <1787, 2>, <73, -4>, <5653, -2> ]
> E;
Elliptic Curve defined by y^2 = x^3 + x^2 - 25*x + 39 over Rational Field
> factor(1088);
[ <2, 6>, <17, 1> ]
1
>