[was@descent alina]$ [was@descent alina]$ Magma V2.7-1      Tue Aug 15 2000 23:59:18 on descent  [Seed = 966383970]
Type ? for help.  Type <Ctrl>-D to quit.

Loading startup file "/home/was/modsym/init-magma.m"

C               IndexGamma0     R               factormod       padiccharpoly
CS              MS              Tn              fcp             qexp
DC              ND              Z               fn              x
ES              NS              charpoly        idxG0
F               Q               ellap           modcharpoly
> EllipticCurve(CremonaDatabase(),"11A");
Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
> aInvariants($1);
[ 0, -1, 1, -10, -20 ]
> EllipticCurve(CremonaDatabase(),"37A");
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
> aInvariants($1);
[ 0, 0, 1, -1, 0 ]
> E:=EllipticCurve(CremonaDatabase(),"37A");
> BaseExtend(E,GF(3));
Elliptic Curve defined by y^2 + y = x^3 + 2*x over GF(3)
> IsCyclic(MordellWeilGroup(BaseExtend(E,GF(3))));
> IsCyclic(MordellWeilGroup(BaseExtend(E,GF(3))));

>> IsCyclic(MordellWeilGroup(
                            ^
Runtime error in 'MordellWeilGroup': Curve must be defined over the rational field
> IsCyclic(TorsionSubgroup(BaseExtend(E,GF(3))));
> IsCyclic(TorsionSubgroup(BaseExtend(E,GF(3))));
true
> [<p,IsCyclic(TorsionSubgroup(BaseExtend(E,GF(p)))) : p in [2..37]];

>> [<p,IsCyclic(TorsionSubgroup(BaseExtend(E,GF(p)))) : p in [2..37]];
                                                      ^
User error: bad syntax
>  [<p,IsCyclic(TorsionSubgroup(BaseExtend(E,GF(p))))> : p in [2..37]];

>>  [<p,IsCyclic(TorsionSubgroup(BaseExtend(E,GF(p))))> : p in [2..37]];
                                                ^
Runtime error in 'GF': Argument 1 (6) is not a prime power
>  [<p,IsCyclic(TorsionSubgroup(BaseExtend(E,GF(p))))> : p in [2..37] |IsPrime(p)];

>>  [<p,IsCyclic(TorsionSubgroup(BaseExtend(E,GF(p))))> : p in [2..37] |IsPrim
                                           ^
Runtime error in 'BaseExtend': Curve is singular
>   [<p,IsCyclic(TorsionSubgroup(BaseExtend(E,GF(p))))> : p in [2..37] |IsPrime(p) and p ne 37];
[ <2, true>, <3, true>, <5, true>, <7, true>, <11, true>, <13, true>, <17, true>, <19, true>, <23, true>, <29, true>, <31, true> ]
>   [<p,IsCyclic(TorsionSubgroup(BaseExtend(E,GF(p))))> : p in [2..97] |IsPrime(p) and p ne 37];
[ <2, true>, <3, true>, <5, true>, <7, true>, <11, true>, <13, true>, <17, true>, <19, true>, <23, true>, <29, true>, <31, true>, <41, true>, <43, true>, <47, true>, <53, true>, <59, true>, <61, true>, <67, false>, <71, true>, <73, true>, <79, true>, <83, true>, <89, true>, <97, true> ]
>  for p in [n : n in [2..97] | IsPrime(n) and n ne 37] do <p,IsCyclic(TorsionSubgroup(BaseExtend(E,GF(p))))>; end for;
<2, true>
<3, true>
<5, true>
<7, true>
<11, true>
<13, true>
<17, true>
<19, true>
<23, true>
<29, true>
<31, true>
<41, true>
<43, true>
<47, true>
<53, true>
<59, true>
<61, true>
<67, false>
<71, true>
<73, true>
<79, true>
<83, true>
<89, true>
<97, true>
>  for p in [n : n in [98..300] | IsPrime(n) and n ne 37] do <p,IsCyclic(TorsionSubgroup(BaseExtend(E,GF(p))))>; end for;
<101, true>
<103, true>
<107, false>
<109, true>
<113, true>
<127, true>
<131, true>
<137, false>
<139, false>
<149, true>
<151, false>
<157, true>
<163, true>
<167, true>
<173, true>
<179, true>
<181, true>
<191, true>
<193, true>
<197, true>
<199, true>
<211, true>
<223, true>
<227, true>
<229, true>
<233, false>
<239, true>
<241, true>
<251, true>
<257, true>
<263, true>
<269, false>
<271, true>
<277, true>
<281, true>
<283, true>
<293, false>
> V:=[<p,IsCyclic(TorsionSubgroup(BaseExtend(E,GF(p))))> : p in [2..300] |IsPrime(p) and p ne 37];
> V;
[ <2, true>, <3, true>, <5, true>, <7, true>, <11, true>, <13, true>, <17, true>, <19, true>, <23, true>, <29, true>, <31, true>, <41, true>, <43, true>, <47, true>, <53, true>, <59, true>, <61, true>, <67, false>, <71, true>, <73, true>, <79, true>, <83, true>, <89, true>, <97, true>, <101, true>, <103, true>, <107, false>, <109, true>, <113, true>, <127, true>, <131, true>, <137, false>, <139, false>, <149, true>, <151, false>, <157, true>, <163, true>, <167, true>, <173, true>, <179, true>, <181, true>, <191, true>, <193, true>, <197, true>, <199, true>, <211, true>, <223, true>, <227, true>, <229, true>, <233, false>, <239, true>, <241, true>, <251, true>, <257, true>, <263, true>, <269, false>, <271, true>, <277, true>, <281, true>, <283, true>, <293, false> ]
> a:=[v : v in V | v[2] eq false];
> a;
[ <67, false>, <107, false>, <137, false>, <139, false>, <151, false>, <233, false>, <269, false>, <293, false> ]
> a:=[v[1] : v in V | v[2] eq false];
> time V:=[<p,IsCyclic(TorsionSubgroup(BaseExtend(E,GF(p))))> : p in [2..997] |IsPrime(p) and p ne 37];
Time: 3.879
> V;
[ <2, true>, <3, true>, <5, true>, <7, true>, <11, true>, <13, true>, <17, true>, <19, true>, <23, true>, <29, true>, <31, true>, <41, true>, <43, true>, <47, true>, <53, true>, <59, true>, <61, true>, <67, false>, <71, true>, <73, true>, <79, true>, <83, true>, <89, true>, <97, true>, <101, true>, <103, true>, <107, false>, <109, true>, <113, true>, <127, true>, <131, true>, <137, false>, <139, false>, <149, true>, <151, false>, <157, true>, <163, true>, <167, true>, <173, true>, <179, true>, <181, true>, <191, true>, <193, true>, <197, true>, <199, true>, <211, true>, <223, true>, <227, true>, <229, true>, <233, false>, <239, true>, <241, true>, <251, true>, <257, true>, <263, true>, <269, false>, <271, true>, <277, true>, <281, true>, <283, true>, <293, false>, <307, true>, <311, true>, <313, true>, <317, false>, <331, true>, <337, true>, <347, true>, <349, false>, <353, true>, <359, true>, <367, false>, <373, true>, <379, true>, <383, true>, <389, true>, <397, true>, <401, true>, <409, true>, <419, true>, <421, true>, <431, true>, <433, true>, <439, true>, <443, true>, <449, true>, <457, true>, <461, true>, <463, true>, <467, true>, <479, true>, <487, true>, <491, false>, <499, true>, <503, true>, <509, true>, <521, true>, <523, true>, <541, true>, <547, true>, <557, true>, <563, true>, <569, true>, <571, true>, <577, true>, <587, true>, <593, true>, <599, true>, <601, false>, <607, true>, <613, true>, <617, true>, <619, true>, <631, true>, <641, true>, <643, true>, <647, true>, <653, true>, <659, true>, <661, true>, <673, true>, <677, true>, <683, true>, <691, false>, <701, true>, <709, true>, <719, true>, <727, true>, <733, true>, <739, true>, <743, true>, <751, true>, <757, true>, <761, true>, <769, true>, <773, true>, <787, true>, <797, true>, <809, true>, <811, false>, <821, true>, <823, false>, <827, true>, <829, true>, <839, false>, <853, false>, <857, true>, <859, true>, <863, false>, <877, false>, <881, false>, <883, true>, <887, true>, <907, true>, <911, true>, <919, true>, <929, false>, <937, true>, <941, false>, <947, true>, <953, true>, <967, true>, <971, false>, <977, true>, <983, true>, <991, true>, <997, true> ]
> W:=[v[1] : v in V | v[2] eq false];
> W;
[ 67, 107, 137, 139, 151, 233, 269, 293, 317, 349, 367, 491, 601, 691, 811, 823, 839, 853, 863, 877, 881, 929, 941, 971 ]
> #W;
24
> #V;
167
> 24/167.0
> ;
0.1437125748502994011976047904
> time V:=V cat [<p,IsCyclic(TorsionSubgroup(BaseExtend(E,GF(p))))> : p in [998..5000] |IsPrime(p) and p ne 37];
Time: 25.070
> W:=[v[1] : v in V | v[2] eq false];
> #W/#V*1.0;
0.1691616766467065868263473053
> W;
[ 67, 107, 137, 139, 151, 233, 269, 293, 317, 349, 367, 491, 601, 691, 811, 823, 839, 853, 863, 877, 881, 929, 941, 971, 1061, 1069, 1103, 1129, 1163, 1181, 1237, 1259, 1279, 1283, 1307, 1373, 1423, 1433, 1489, 1553, 1601, 1607, 1627, 1669, 1693, 1777, 1783, 1787, 1847, 1877, 1973, 2143, 2179, 2213, 2267, 2287, 2371, 2389, 2393, 2543, 2549, 2579, 2593, 2657, 2713, 2741, 2801, 2803, 2861, 2879, 2897, 2927, 2953, 2963, 2971, 3119, 3181, 3257, 3259, 3319, 3371, 3391, 3511, 3623, 3637, 3691, 3697, 3701, 3709, 3727, 3821, 3847, 4003, 4099, 4153, 4217, 4219, 4229, 4259, 4271, 4441, 4481, 4567, 4597, 4603, 4651, 4657, 4703, 4799, 4909, 4957, 4967, 4999 ]
> time V:=V cat [<p,IsCyclic(TorsionSubgroup(BaseExtend(E,GF(p))))> : p in [5001..10000] |IsPrime(p) and p ne 37];
Time: 26.730
> W:=[v[1] : v in V | v[2] eq false];
> #W/#V*1.0;
0.1710097719869706840390879478
> E67,f:=BaseExtend(E,GF(67));

>> E67,f:=BaseExtend(E,GF(67));
        ^
Runtime error in :=: Expected to assign 2 value(s) but only computed 1 value(s)
> G,f:=MordellWeilGroup(E);
> f(G.1);
(0 : 0 : 1)
> S:=[<p,IsCyclic(TorsionSubgroup(Ep)),Order(TorsionSubgroup(Ep)),Order(Ep![0,0])> : p in [1..1000] |IsPrime(p) and p ne 37 where Ep:=BaseExtend(E,GF(p))];
> S:=[<p,IsCyclic(TorsionSubgroup(Ep)),Order(TorsionSubgroup(Ep)),Order(Ep![0,0])> : p in [1..1000] |IsPrime(p) and p ne 37 where Ep:=BaseExtend(E,GF(p))];

>> and p ne 37 where Ep:=BaseExtend(E,GF(p))];
                                        ^
Runtime error in 'GF': Argument 1 (1) should be >= 2
> S:=[<p,IsCyclic(TorsionSubgroup(Ep)),Order(TorsionSubgroup(Ep)),Order(Ep![0,0])> where Ep:=BaseExtend(E,GF(p)) : p in [1..1000] |IsPrime(p) and p ne 37 ];
> S;
[ <2, true, 5, 5>, <3, true, 7, 7>, <5, true, 8, 8>, <7, true, 9, 9>, <11, true, 17, 17>, <13, true, 16, 16>, <17, true, 18, 18>, <19, true, 20, 20>, <23, true, 22, 11>, <29, true, 24, 12>, <31, true, 36, 18>, <41, true, 51, 51>, <43, true, 42, 14>, <47, true, 57, 57>, <53, true, 53, 53>, <59, true, 52, 13>, <61, true, 70, 35>, <67, false, 60, 30>, <71, true, 63, 63>, <73, true, 75, 75>, <79, true, 76, 76>, <83, true, 99, 33>, <89, true, 86, 86>, <97, true, 94, 47>, <101, true, 99, 33>, <103, true, 86, 86>, <107, false, 120, 20>, <109, true, 126, 126>, <113, true, 132, 33>, <127, true, 127, 127>, <131, true, 144, 144>, <137, false, 144, 72>, <139, false, 136, 17>, <149, true, 155, 155>, <151, false, 136, 68>, <157, true, 135, 15>, <163, true, 182, 182>, <167, true, 180, 36>, <173, true, 165, 55>, <179, true, 162, 81>, <181, true, 177, 177>, <191, true, 196, 49>, <193, true, 220, 220>, <197, true, 195, 39>, <199, true, 198, 99>, <211, true, 225, 45>, <223, true, 241, 241>, <227, true, 244, 122>, <229, true, 223, 223>, <233, false, 228, 38>, <239, true, 246, 123>, <241, true, 228, 114>, <251, true, 254, 254>, <257, true, 258, 129>, <263, true, 245, 49>, <269, false, 276, 138>, <271, true, 303, 303>, <277, true, 266, 133>, <281, true, 270, 135>, <283, true, 280, 28>, <293, false, 296, 74>, <307, true, 325, 325>, <311, true, 312, 104>, <313, true, 292, 292>, <317, false, 296, 148>, <331, true, 334, 167>, <337, true, 363, 121>, <347, true, 358, 179>, <349, false, 344, 172>, <353, true, 346, 346>, <359, true, 375, 375>, <367, false, 360, 180>, <373, true, 393, 393>, <379, true, 365, 73>, <383, true, 364, 91>, <389, true, 386, 193>, <397, true, 403, 403>, <401, true, 384, 384>, <409, true, 390, 195>, <419, true, 413, 413>, <421, true, 446, 446>, <431, true, 462, 154>, <433, true, 425, 425>, <439, true, 412, 103>, <443, true, 443, 443>, <449, true, 414, 414>, <457, true, 440, 440>, <461, true, 432, 432>, <463, true, 486, 243>, <467, true, 470, 47>, <479, true, 466, 233>, <487, true, 512, 512>, <491, false, 520, 260>, <499, true, 488, 61>, <503, true, 488, 488>, <509, true, 541, 541>, <521, true, 555, 555>, <523, true, 546, 546>, <541, true, 522, 261>, <547, true, 540, 270>, <557, true, 576, 144>, <563, true, 594, 594>, <569, true, 594, 297>, <571, true, 565, 565>, <577, true, 578, 578>, <587, true, 620, 620>, <593, true, 599, 599>, <599, true, 599, 599>, <601, false, 624, 156>, <607, true, 640, 640>, <613, true, 599, 599>, <617, true, 601, 601>, <619, true, 621, 207>, <631, true, 660, 660>, <641, true, 643, 643>, <643, true, 630, 315>, <647, true, 656, 656>, <653, true, 678, 339>, <659, true, 675, 135>, <661, true, 690, 690>, <673, true, 647, 647>, <677, true, 689, 689>, <683, true, 666, 74>, <691, false, 712, 178>, <701, true, 714, 119>, <709, true, 670, 134>, <719, true, 681, 227>, <727, true, 712, 712>, <733, true, 727, 727>, <739, true, 749, 749>, <743, true, 723, 723>, <751, true, 727, 727>, <757, true, 808, 202>, <761, true, 797, 797>, <769, true, 744, 93>, <773, true, 783, 783>, <787, true, 793, 793>, <797, true, 746, 373>, <809, true, 808, 404>, <811, false, 765, 255>, <821, true, 869, 869>, <823, false, 840, 35>, <827, true, 806, 403>, <829, true, 834, 417>, <839, false, 796, 398>, <853, false, 828, 138>, <857, true, 906, 302>, <859, true, 880, 880>, <863, false, 888, 444>, <877, false, 828, 414>, <881, false, 896, 448>, <883, true, 836, 209>, <887, true, 863, 863>, <907, true, 856, 428>, <911, true, 886, 886>, <919, true, 978, 978>, <929, false, 912, 228>, <937, true, 901, 53>, <941, false, 952, 476>, <947, true, 936, 936>, <953, true, 893, 893>, <967, true, 982, 491>, <971, false, 980, 490>, <977, true, 950, 475>, <983, true, 975, 25>, <991, true, 1010, 202>, <997, true, 1040, 520> ]
> T:=[s : s in S | s[3] eq s[4]];
> #S;
167
> #T;
76
> #T/#S;
76/167
> #T/#S * 1.0;
0.4550898203592814371257485029
> S:=S cat [<p,IsCyclic(TorsionSubgroup(Ep)),Order(TorsionSubgroup(Ep)),Order(Ep![0,0])> where Ep:=BaseExtend(E,GF(p)) : p in [1001..5000] |IsPrime(p) and p ne 37 ];
> T:=[s : s in S | s[3] eq s[4]];
> #T/#S * 1.0;
0.4655688622754491017964071856
> [t[4] : t in T];
[ 5, 7, 8, 9, 17, 16, 18, 20, 51, 57, 53, 63, 75, 76, 86, 86, 126, 127, 144, 155, 182, 177, 220, 241, 223, 254, 303, 325, 292, 346, 375, 393, 403, 384, 413, 446, 425, 443, 414, 440, 432, 512, 488, 541, 555, 546, 594, 565, 578, 620, 599, 599, 640, 599, 601, 660, 643, 656, 690, 647, 689, 712, 727, 749, 723, 727, 797, 783, 793, 869, 880, 863, 886, 978, 936, 893, 1057, 978, 974, 1084, 1031, 1099, 1057, 1145, 1177, 1136, 1221, 1205, 1158, 1226, 1259, 1278, 1263, 1242, 1257, 1336, 1272, 1324, 1287, 1330, 1296, 1319, 1425, 1411, 1459, 1454, 1493, 1505, 1430, 1497, 1473, 1469, 1523, 1491, 1566, 1542, 1516, 1564, 1549, 1570, 1668, 1605, 1585, 1720, 1713, 1762, 1729, 1674, 1738, 1806, 1799, 1833, 1781, 1889, 1824, 1966, 1923, 1981, 1931, 2053, 1921, 1961, 2098, 2092, 2135, 2029, 2153, 2093, 2054, 2155, 2157, 2172, 2075, 2197, 2162, 2193, 2270, 2202, 2244, 2200, 2341, 2283, 2333, 2287, 2352, 2384, 2281, 2359, 2316, 2311, 2407, 2460, 2371, 2470, 2409, 2492, 2450, 2554, 2570, 2521, 2553, 2600, 2665, 2584, 2569, 2708, 2598, 2691, 2652, 2789, 2781, 2809, 2855, 2859, 2798, 2827, 2846, 2903, 3051, 2959, 3113, 2983, 3135, 3139, 3143, 3074, 3045, 3146, 3194, 3211, 3213, 3165, 3174, 3265, 3249, 3226, 3363, 3317, 3284, 3449, 3393, 3430, 3523, 3442, 3439, 3537, 3514, 3643, 3486, 3505, 3614, 3569, 3525, 3573, 3660, 3748, 3667, 3737, 3782, 3807, 3866, 3921, 3886, 3905, 3892, 3901, 3992, 3991, 3933, 3973, 3861, 3933, 3989, 4104, 3991, 3926, 4039, 4144, 3977, 4109, 4061, 4222, 4137, 4209, 4320, 4264, 4345, 4400, 4427, 4321, 4433, 4391, 4425, 4488, 4374, 4483, 4427, 4488, 4581, 4613, 4471, 4543, 4516, 4477, 4501, 4454, 4619, 4525, 4627, 4556, 4706, 4587, 4642, 4783, 4689, 4638, 4672, 4871, 4902, 4887, 4837, 4763, 4847, 4942, 4881, 4877, 4901, 4903, 5073, 4906, 4972 ]
> f(2*G.1);
(1 : 0 : 1)
> S2:=[<p,IsCyclic(TorsionSubgroup(Ep)),Order(TorsionSubgroup(Ep)),Order(Ep![0,0])> where Ep:=BaseExtend(E,GF(p)) : p in [1..1000] |IsPrime(p) and p ne 37 ];
> T2:=[s : s in S | s[3] eq s[4]];
> #T2 / #S2 *1.0;
1.862275449101796407185628742
> #S2 / #T2 *1.0;
0.5369774919614147909967845659
> T2:=[s : s in S2 | s[3] eq s[4]];
> #S2 / #T2 *1.0;
2.197368421052631578947368421
> #T2 / #S2 *1.0;
0.4550898203592814371257485029
> S2:=[<p,IsCyclic(TorsionSubgroup(Ep)),Order(TorsionSubgroup(Ep)),Order(Ep![1,0])> where Ep:=BaseExtend(E,GF(p)) : p in [1..1000] |IsPrime(p) and p ne 37 ];
> T2:=[s : s in S2 | s[3] eq s[4]];
> #T2 / #S2 *1.0;
0.2514970059880239520958083832
> f(3*G.1);
(-1 : -1 : 1)
> S2:=[<p,IsCyclic(TorsionSubgroup(Ep)),Order(TorsionSubgroup(Ep)),Order(Ep![-1,-1])> where Ep:=BaseExtend(E,GF(p)) : p in [1..1000] |IsPrime(p) and p ne 37 ];
> T2:=[s : s in S2 | s[3] eq s[4]];
> #T2 / #S2 *1.0;
0.3113772455089820359281437125
> f(5*G.1);
(1/4 : -5/8 : 1)
> S5:=[<p,IsCyclic(TorsionSubgroup(Ep)),Order(TorsionSubgroup(Ep)),Order(Ep![1/4,-5/8])> where Ep:=BaseExtend(E,GF(p)) : p in [1..1000] |IsPrime(p) and p not in [2,37] ];

>> GF(p)) : p in [1..1000] |IsPrime(p) and p not in [2,37] ];
                                             ^
User error: bad syntax
> S5:=[<p,IsCyclic(TorsionSubgroup(Ep)),Order(TorsionSubgroup(Ep)),Order(Ep![1/4,-5/8])> where Ep:=BaseExtend(E,GF(p)) : p in [1..1000] |IsPrime(p) and p ne 2 and p ne 37 ];
> T5:=[s : s in S5 | s[3] eq s[4]];
> #T5/#S5;
30/83
> $1*1.0;
0.3614457831325301204819277108
> S;

> 
[Interrupt twice in half a second; exiting]

Total time: 105.689 seconds
[was@descent alina]$ exit
exit

Process magma finished
[was@descent alina]$ [was@descent alina]$ Magma V2.7-1      Wed Aug 16 2000 00:57:12 on descent  [Seed = 1442155185]
Type ? for help.  Type <Ctrl>-D to quit.

Loading startup file "/home/was/modsym/init-magma.m"

C               IndexGamma0     R               factormod       padiccharpoly
CS              MS              Tn              fcp             qexp
DC              ND              Z               fn              x
ES              NS              charpoly        idxG0
F               Q               ellap           modcharpoly
> quit;

Total time: 1.929 seconds
[was@descent alina]$ exit
exit

Process magma finished
P=[0,0]  generates 37A

   50%   it generates E(F_p)

   25% of time 2P generates E(F_p)

   31% of time 3P generates E(F_p)

   36% of time 5P generates E(F_p)