Magma V2.18-8     Wed Sep  5 2012 13:11:16 on luna     [Seed = 487226911]
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Loading file "rationalpoints5tuples.mg"
Version 24/7/2012
> Cuenta00({ 0, 1, 2, 5, 7 }, { 2, 5, 7 }, 3, 2);                                                                                                                        

=================================================
=========      { 0, 1, 2, 5, 7 }      =========
=================================================

XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
==== J= { 2, 5, 7 } ; {j1,j2}={3,2} ==== 
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX

D=14 --> deltas=[ 1, -1, 2, -2, 5, -5, -10, 10 ]

::: (1,1)   :::   (1)*(t^2 + (a - 7)*t + 1/2*(-2*a + 9))*(t^2 - 4*t + 5/2)

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

        ***** h_K = 1 --> Minimal Model

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

$$$$$$$$$$ [OK]  --------> [ (1 : 0), (1 : 0) ]

t=oo -> (q,a)=(0,1) --> [ 1, 1, 1, 1, 1 ]

::: (1,1)   :::   (-1)*(t^2 + (a - 7)*t + 1/2*(-2*a + 9))*(t^2 - 4*t + 5/2)

        *** Elliptic curve base point = (1/78*(25*a + 155) : 1/6084*(270*a + 335) : 1)

        ***** h_K = 1 --> Minimal Model

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

$$$$$$$$$$ [OK]  --------> []


::: (1,1)   :::   (2)*(t^2 + (a - 7)*t + 1/2*(-2*a + 9))*(t^2 - 4*t + 5/2)

        *** Elliptic curve base point = (3 : 1/2*(a - 4) : 1)

        ***** h_K = 1 --> Minimal Model

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

#####   Rank = 2 > 1

::: (1,2)   :::   (2)*(t^2 + (a - 7)*t + 1/2*(-2*a + 9))*(t^2 - 8*t + 5/2)

        *** Elliptic curve base point = (3 : 1/2*(-5*a + 20) : 1)

        ***** h_K = 1 --> Minimal Model

          - RankBound for delta = 2 --> r = 1

$$$$$$$$$$ [OK]  --------> [ (3 : 1), (3 : 1) ]

t=3 -> (q,a)=(24,1) --> [ 1, 25, 49, 121, 169 ]

::: (1,1)   :::   (-2)*(t^2 + (a - 7)*t + 1/2*(-2*a + 9))*(t^2 - 4*t + 5/2)

#####   Bruin-Stoll: No Answer--> #Hk=4

::: (1,2)   :::   (-2)*(t^2 + (a - 7)*t + 1/2*(-2*a + 9))*(t^2 - 8*t + 5/2)

        *** Elliptic curve base point = (1/8*(3*a + 14) : 1/32*(-9*a - 72) : 1)

        ***** h_K = 1 --> Minimal Model

          - RankBound for delta = -2 --> r = 1

$$$$$$$$$$ [OK]  --------> []


::: (1,1)   :::   (5)*(t^2 + (a - 7)*t + 1/2*(-2*a + 9))*(t^2 - 4*t + 5/2)

        *** Elliptic curve base point = (-a + 7 : 1/2*(-15*a + 55) : 1)

        ***** h_K = 1 --> Minimal Model

          - RankBound for delta = 5 --> r = 2

#####   Rank = 2 > 1

::: (1,2)   :::   (5)*(t^2 + (a - 7)*t + 1/2*(-2*a + 9))*(t^2 - 8*t + 5/2)

        *** Elliptic curve base point = (5/6 : 1/36*(-25*a - 75) : 1)

        ***** h_K = 1 --> Minimal Model

          - RankBound for delta = 5 --> r = 1

$$$$$$$$$$ [OK]  --------> [ (5/6 : 1), (5/6 : 1) ]

t=5/6 -> (q,a)=(24,1) --> [ 1, 25, 49, 121, 169 ]

::: (1,1)   :::   (-5)*(t^2 + (a - 7)*t + 1/2*(-2*a + 9))*(t^2 - 4*t + 5/2)

#####   Bruin-Stoll: No Answer--> #Hk=4

::: (1,2)   :::   (-5)*(t^2 + (a - 7)*t + 1/2*(-2*a + 9))*(t^2 - 8*t + 5/2)

        *** Elliptic curve base point = (1/4*(-5*a + 20) : 1/8*(15*a - 55) : 1)

        ***** h_K = 1 --> Minimal Model

          - RankBound for delta = -5 --> r = 1

$$$$$$$$$$ [OK]  --------> []


::: (1,1)   :::   (-10)*(t^2 + (a - 7)*t + 1/2*(-2*a + 9))*(t^2 - 4*t + 5/2)

        *** Elliptic curve base point = (1/47*(15*a + 93) : 1/4418*(-275*a - 2410) : 1)

        ***** h_K = 1 --> Minimal Model

          - RankBound for delta = -10 --> r = 1

$$$$$$$$$$ [OK]  --------> []


::: (1,1)   :::   (10)*(t^2 + (a - 7)*t + 1/2*(-2*a + 9))*(t^2 - 4*t + 5/2)

        *** Elliptic curve base point = (0 : 1/2*(-5*a + 10) : 1)

        ***** h_K = 1 --> Minimal Model

          - RankBound for delta = 10 --> r = 1

$$$$$$$$$$ [OK]  --------> [ (0 : 1), (0 : 1) ]

t=0 -> (q,a)=(0,1) --> [ 1, 1, 1, 1, 1 ]

##############
 SOLUTION:
 {
[ 1, 25, 49, 121, 169 ],
[ 1, 1, 1, 1, 1 ]
}
##############
true {
    [ 1, 25, 49, 121, 169 ],
    [ 1, 1, 1, 1, 1 ]
}
{ (3 : 1), (1 : 0), (5/6 : 1), (0 : 1) }
<{ 0, 1, 2, 5, 7 }, { 2, 5, 7 }, [ 3, 2 ], 14, [ 1, -1, 2, -2, 5, -5, -10, 10 ], <1, <[ 1, 1 ], t^2 + (a - 7)*t + 1/2*(-2*a + 9), t^2 - 4*t + 5/2>, <<1, -1, 0>, <0, a + 
    1, 0, -7158*a - 26781, 625809*a + 2341563>, <(1/2*(36*a + 139) : 1/4*(-37*a - 134) : 1), (17*a + 65 : 0 : 1)>, 1, <(1 : 0)>, <[ 1, 1, 1, 1, 1 ]>>>, <-1, <[ 1, 1 ], 
    t^2 + (a - 7)*t + 1/2*(-2*a + 9), t^2 - 4*t + 5/2>, <<1/78*(25*a + 155), 1/6084*(270*a + 335), 1>, <0, -a - 1, 0, -7158*a - 26781, -625809*a - 2341563>, 
<(1/92450*(7421568*a + 27835919) : 1/39753500*(-71057351807*a - 265940857106) : 1), (-17*a - 65 : 0 : 1)>, 1, <>, <>>>, <2, <[ 1, 2 ], t^2 + (a - 7)*t + 1/2*(-2*a + 9), 
    t^2 - 8*t + 5/2>, <<3, 1/2*(-5*a + 20), 1>, <0, 0, 0, 708*a - 3186, -22264*a + 88088>, <(1/50*(18634*a + 71985) : 1/500*(-10080973*a - 37718022) : 1), (-4*a + 28 : 0 
: 1)>, 1, <(3 : 1)>, <[ 1, 25, 49, 121, 169 ]>>>, <-2, <[ 1, 2 ], t^2 + (a - 7)*t + 1/2*(-2*a + 9), t^2 - 8*t + 5/2>, <<1/8*(3*a + 14), 1/32*(-9*a - 72), 1>, <0, 0, 0, 
    708*a - 3186, 22264*a - 88088>, <(1/2*(-190*a + 953) : 1/4*(-12073*a + 50394) : 1), (4*a - 28 : 0 : 1)>, 1, <>, <>>>, <5, <[ 1, 2 ], t^2 + (a - 7)*t + 1/2*(-2*a + 9),
t^2 - 8*t + 5/2>, <<5/6, 1/36*(-25*a - 75), 1>, <0, 0, 0, -708*a - 3186, 22264*a + 88088>, <(1/50*(-18634*a + 71985) : 1/500*(-10080973*a + 37718022) : 1), (4*a + 28 : 0 
: 1)>, 1, <(5/6 : 1)>, <[ 1, 25, 49, 121, 169 ]>>>, <-5, <[ 1, 2 ], t^2 + (a - 7)*t + 1/2*(-2*a + 9), t^2 - 8*t + 5/2>, <<1/4*(-5*a + 20), 1/8*(15*a - 55), 1>, <0, 0, 0, 
-708*a - 3186, -22264*a - 88088>, <(1/2*(190*a + 953) : 1/4*(-12073*a - 50394) : 1), (-4*a - 28 : 0 : 1)>, 1, <>, <>>>, <-10, <[ 1, 1 ], t^2 + (a - 7)*t + 1/2*(-2*a + 9),
t^2 - 4*t + 5/2>, <<1/47*(15*a + 93), 1/4418*(-275*a - 2410), 1>, <0, a - 1, 0, 7158*a - 26781, 625809*a - 2341563>, <(1/92450*(-7421568*a + 27835919) : 
1/39753500*(-71057351807*a + 265940857106) : 1), (17*a - 65 : 0 : 1)>, 1, <>, <>>>, <10, <[ 1, 1 ], t^2 + (a - 7)*t + 1/2*(-2*a + 9), t^2 - 4*t + 5/2>, <<0, 1/2*(-5*a + 
    10), 1>, <0, -a + 1, 0, 7158*a - 26781, -625809*a + 2341563>, <(1/2*(-36*a + 139) : 1/4*(-37*a + 134) : 1), (-17*a + 65 : 0 : 1)>, 1, <(0 : 1)>, <[ 1, 1, 1, 1, 1 
]>>>>
>