Magma V2.18-8     Wed Sep  5 2012 13:11:35 on luna     [Seed = 3519460064]
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Loading file "rationalpoints5tuples.mg"
Version 24/7/2012
> 
> Cuenta00({ 0, 13, 24, 33, 49 }, { 0, 13, 24 },2,2);                                                                                                                    

=================================================
=========      { 0, 13, 24, 33, 49 }      =========
=================================================

XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
==== J= { 0, 13, 24 } ; {j1,j2}={2,2} ==== 
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX

D=165 --> deltas=[ 1, 6, 10, 11, 14, 21, 35, 154 ]

::: (1,1)   :::   (1)*(t^2 + 1/11*(4*a + 18)*t + 24/11)*(t^2 + 134/11*t + 24/11)

        *** Elliptic curve base point = (1 : -1 : 0)

        ***** h_K = 2 --> Integral Model

          - RankBound for delta = 1 --> r = 1

$$$$$$$$$$ [OK]  --------> [ (1 : 0), (-2/11 : 1), (1 : 0), (0 : 1), (-12 : 1), (0 : 1) ]

t=-2/11 -> (q,a)=(-1,49) --> [ 49, 36, 25, 16, 0 ]
t=-12 -> (q,a)=(-1,49) --> [ 49, 36, 25, 16, 0 ]
t=0 -> (q,a)=(0,1) --> [ 1, 1, 1, 1, 1 ]
t=oo -> (q,a)=(0,1) --> [ 1, 1, 1, 1, 1 ]

::: (1,1)   :::   (6)*(t^2 + 1/11*(4*a + 18)*t + 24/11)*(t^2 + 134/11*t + 24/11)

        *** Elliptic curve base point = (-12 : 0 : 1)

        ***** h_K = 2 --> Integral Model

          - RankBound for delta = 6 --> r = 1

#####   PseudoMordellWeilGroup false ===> We try with DescentInformation

Torsion Subgroup = Z/2
The 2-Selmer group has rank 2
After 2-descent: 
    0 <= Rank(E) <= 1
    Sha(E)[2] <= (Z/2)^1
(Searched up to height 16 on the 2-coverings.)
Found a point of infinite order.
After more searching:
    1 <= Rank(E) <= 1
    Sha(E)[2] is trivial
(Searched up to height 62 on the 2-coverings.)


$$$$$$$$$$ [OK]  --------> [ (-12 : 1), (-2/11 : 1) ]

t=-2/11 -> (q,a)=(-1,49) --> [ 49, 36, 25, 16, 0 ]
t=-12 -> (q,a)=(-1,49) --> [ 49, 36, 25, 16, 0 ]

::: (1,1)   :::   (10)*(x^2 + 1/11*(4*a + 18)*x + 24/11)*(x^2 + 134/11*x + 24/11)

        *** Elliptic curve base point = (-12 : 0 : 1)

        ***** h_K = 2 --> Integral Model

          - RankBound for delta = 10 --> r = 2

#####   Rank = 2 > 1

::: (1,2)   :::   (10)*(t^2 + 1/11*(4*a + 18)*t + 24/11)*(t^2 - 34/11*t + 24/11)

        *** Elliptic curve base point = (12/11 : 0 : 1)

        ***** h_K = 2 --> Integral Model

          - RankBound for delta = 10 --> r = 0

$$$$$$$$$$ [OK]  --------> { (2 : 1), (12/11 : 1) }

t=2 -> (q,a)=(-1,49) --> [ 49, 36, 25, 16, 0 ]
t=12/11 -> (q,a)=(-1,49) --> [ 49, 36, 25, 16, 0 ]

::: (1,1)   :::   (11)*(t^2 + 1/11*(4*a + 18)*t + 24/11)*(t^2 + 134/11*t + 24/11)

        *** Elliptic curve base point = (-12 : 0 : 1)

        ***** h_K = 2 --> Integral Model

          - RankBound for delta = 11 --> r = 2

#####   Rank = 2 > 1

::: (1,2)   :::   (11)*(t^2 + 1/11*(4*a + 18)*t + 24/11)*(t^2 - 34/11*t + 24/11)

        *** Elliptic curve base point = (12/11 : 0 : 1)

        ***** h_K = 2 --> Integral Model

          - RankBound for delta = 11 --> r = 0

$$$$$$$$$$ [OK]  --------> { (2 : 1), (12/11 : 1) }

t=2 -> (q,a)=(-1,49) --> [ 49, 36, 25, 16, 0 ]
t=12/11 -> (q,a)=(-1,49) --> [ 49, 36, 25, 16, 0 ]

::: (1,1)   :::   (14)*(t^2 + 1/11*(4*a + 18)*t + 24/11)*(t^2 + 134/11*t + 24/11)

        *** Elliptic curve base point = (-12 : 0 : 1)

        ***** h_K = 2 --> Integral Model

          - RankBound for delta = 14 --> r = 1

#####   PseudoMordellWeilGroup false ===> We try with DescentInformation

Torsion Subgroup = Z/2
The 2-Selmer group has rank 2
After 2-descent: 
    0 <= Rank(E) <= 1
    Sha(E)[2] <= (Z/2)^1
(Searched up to height 16 on the 2-coverings.)
After more searching:
    0 <= Rank(E) <= 1
    Sha(E)[2] <= (Z/2)^1
(Searched up to height 251 on the 2-coverings.)


#####   PseudoMordellWeilGroup and DescentInformation false

::: (1,2)   :::   (14)*(x^2 + 1/11*(4*a + 18)*x + 24/11)*(x^2 - 34/11*x + 24/11)

        *** Elliptic curve base point = (12/11 : 0 : 1)

        ***** h_K = 2 --> Integral Model

          - RankBound for delta = 14 --> r = 1

#####   PseudoMordellWeilGroup false ===> We try with DescentInformation

Torsion Subgroup = Z/2
The 2-Selmer group has rank 2
Found a point of infinite order.
After 2-descent: 
    1 <= Rank(E) <= 1
    Sha(E)[2] is trivial
(Searched up to height 16 on the 2-coverings.)


$$$$$$$$$$ [OK]  --------> [ (12/11 : 1), (2 : 1) ]

t=2 -> (q,a)=(-1,49) --> [ 49, 36, 25, 16, 0 ]
t=12/11 -> (q,a)=(-1,49) --> [ 49, 36, 25, 16, 0 ]

::: (1,1)   :::   (21)*(x^2 + 1/11*(4*a + 18)*x + 24/11)*(x^2 + 134/11*x + 24/11)

        *** Elliptic curve base point = (-12 : 0 : 1)

        ***** h_K = 2 --> Integral Model

          - RankBound for delta = 21 --> r = 1

#####   PseudoMordellWeilGroup false ===> We try with DescentInformation

Torsion Subgroup = Z/2
The 2-Selmer group has rank 2
Found a point of infinite order.
After 2-descent: 
    1 <= Rank(E) <= 1
    Sha(E)[2] is trivial
(Searched up to height 16 on the 2-coverings.)


$$$$$$$$$$ [OK]  --------> [ (-12 : 1), (16/3 : 1), (9/22 : 1), (-2/11 : 1), (16/3 : 1), (9/22 : 1) ]

t=-2/11 -> (q,a)=(-1,49) --> [ 49, 36, 25, 16, 0 ]
t=9/22 -> (q,a)=(24,49) --> [ 49, 361, 625, 841, 1225 ]
t=16/3 -> (q,a)=(24,49) --> [ 49, 361, 625, 841, 1225 ]
t=-12 -> (q,a)=(-1,49) --> [ 49, 36, 25, 16, 0 ]

::: (1,1)   :::   (35)*(x^2 + 1/11*(4*a + 18)*x + 24/11)*(x^2 + 134/11*x + 24/11)

        *** Elliptic curve base point = (-12 : 0 : 1)

        ***** h_K = 2 --> Integral Model

          - RankBound for delta = 35 --> r = 2

#####   Rank = 2 > 1

::: (1,2)   :::   (35)*(t^2 + 1/11*(4*a + 18)*t + 24/11)*(t^2 - 34/11*t + 24/11)

        *** Elliptic curve base point = (12/11 : 0 : 1)

        ***** h_K = 2 --> Integral Model

          - RankBound for delta = 35 --> r = 0

$$$$$$$$$$ [OK]  --------> { (2 : 1), (12/11 : 1) }

t=2 -> (q,a)=(-1,49) --> [ 49, 36, 25, 16, 0 ]
t=12/11 -> (q,a)=(-1,49) --> [ 49, 36, 25, 16, 0 ]

::: (1,1)   :::   (154)*(t^2 + 1/11*(4*a + 18)*t + 24/11)*(t^2 + 134/11*t + 24/11)

        *** Elliptic curve base point = (-12 : 0 : 1)

        ***** h_K = 2 --> Integral Model

          - RankBound for delta = 154 --> r = 2

#####   Rank = 2 > 1

::: (1,2)   :::   (154)*(t^2 + 1/11*(4*a + 18)*t + 24/11)*(t^2 - 34/11*t + 24/11)

        *** Elliptic curve base point = (12/11 : 0 : 1)

        ***** h_K = 2 --> Integral Model

          - RankBound for delta = 154 --> r = 0

$$$$$$$$$$ [OK]  --------> { (2 : 1), (12/11 : 1) }

t=2 -> (q,a)=(-1,49) --> [ 49, 36, 25, 16, 0 ]
t=12/11 -> (q,a)=(-1,49) --> [ 49, 36, 25, 16, 0 ]

##############
 SOLUTION:
 {
[ 49, 36, 25, 16, 0 ],
[ 49, 361, 625, 841, 1225 ],
[ 1, 1, 1, 1, 1 ]
}
##############
true {
    [ 49, 36, 25, 16, 0 ],
    [ 49, 361, 625, 841, 1225 ],
    [ 1, 1, 1, 1, 1 ]
}
{ (2 : 1), (12/11 : 1), (16/3 : 1), (9/22 : 1), (1 : 0), (-2/11 : 1), (0 : 1), (-12 : 1) }
<{ 0, 13, 24, 33, 49 }, { 0, 13, 24 }, [ 2, 2 ], 165, [ 1, 6, 10, 11, 14, 21, 35, 154 ], <1, <[ 1, 1 ], t^2 + 1/11*(4*a + 18)*t + 24/11, t^2 + 134/11*t + 24/11>, <<1, -1,
0>, <4*a + 152, -464*a + 6992, -19168*a + 458528, -1622144*a + 20824192, 0>, <(224*a - 3792 : -896*a + 15168 : 1), (232*a - 4024 : 0 : 1)>, 1, <(1 : 0), (-2/11 : 1), (0 :
1), (-12 : 1)>, <[ 49, 36, 25, 16, 0 ], [ 1, 1, 1, 1, 1 ]>>>, <6, <[ 1, 1 ], t^2 + 1/11*(4*a + 18)*t + 24/11, t^2 + 134/11*t + 24/11>, <<-12, 0, 1>, <0, 371712*a + 
    1489752, 0, -15348453120*a + 1927327184640, -24005156090572800*a + 416365293570969600>, <(1/2209*(-157367760*a + 2984593260) : 1/103823*(8146344345480*a + 
    47105382986640) : 1), (5808*a - 168432 : 0 : 1)>, 1, <(-2/11 : 1), (-12 : 1)>, <[ 49, 36, 25, 16, 0 ]>>>, <10, <[ 1, 2 ], t^2 + 1/11*(4*a + 18)*t + 24/11, t^2 - 
    34/11*t + 24/11>, <<12/11, 0, 1>, <0, 154880*a + 2134440, 0, 159762592000*a + 2056299168000, 35669317798400000*a + 458213544025600000>, "Rank 0", 0, <(2 : 1), (12/11 
: 1)>, <[ 49, 36, 25, 16, 0 ]>>>, <11, <[ 1, 2 ], t^2 + 1/11*(4*a + 18)*t + 24/11, t^2 - 34/11*t + 24/11>, <<12/11, 0, 1>, <0, 1408*a + 19404, 0, 13203520*a + 169942080, 
26798886400*a + 344262617600>, "Rank 0", 0, <(2 : 1), (12/11 : 1)>, <[ 49, 36, 25, 16, 0 ]>>>, <14, <[ 1, 2 ], t^2 + 1/11*(4*a + 18)*t + 24/11, t^2 - 34/11*t + 24/11>, 
<<12/11, 0, 1>, <0, 216832*a + 2988216, 0, 313134680320*a + 4030346369280, 97876608038809600*a + 1257337964806246400>, <(1/75*(-4402464*a - 58446872) : 
1/1125*(-16160639968*a - 207549878976) : 1), (-149072*a - 1937936 : 0 : 1)>, 1, <(2 : 1), (12/11 : 1)>, <[ 49, 36, 25, 16, 0 ]>>>, <21, <[ 1, 1 ], t^2 + 1/11*(4*a + 18)*t
    + 24/11, t^2 + 134/11*t + 24/11>, <<-12, 0, 1>, <0, 1300992*a + 5214132, 0, -188018550720*a + 23609758011840, -1029221067383308800*a + 17851661961855321600>, 
<(-406560*a + 11790240 : -328703760*a + 41275800720 : 1), (20328*a - 589512 : 0 : 1)>, 1, <(16/3 : 1), (9/22 : 1), (-2/11 : 1), (-12 : 1)>, <[ 49, 36, 25, 16, 0 ], [ 49, 
361, 625, 841, 1225 ]>>>, <35, <[ 1, 2 ], t^2 + 1/11*(4*a + 18)*t + 24/11, t^2 - 34/11*t + 24/11>, <<12/11, 0, 1>, <0, 542080*a + 7470540, 0, 1957091752000*a + 
    25189664808000, 1529322000606400000*a + 19645905700097600000>, "Rank 0", 0, <(2 : 1), (12/11 : 1)>, <[ 49, 36, 25, 16, 0 ]>>>, <154, <[ 1, 2 ], t^2 + 1/11*(4*a + 
    18)*t + 24/11, t^2 - 34/11*t + 24/11>, <<12/11, 0, 1>, <0, 19712*a + 271656, 0, 2587889920*a + 33308647680, 73536144281600*a + 944656622694400>, "Rank 0", 0, <(2 : 
1), (12/11 : 1)>, <[ 49, 36, 25, 16, 0 ]>>>>
>