kummer-notes-code

Notes and code from my early research in Kummer Theory for Elliptic Curves.
git clone https://git.tronto.net/kummer-notes-code
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bunch_of_curves.gp (38893B)


      1 
      2 \\ Elliptic curves downloaded from the LMFDB downloaded on 31 January 2019.
      3 \\ Below is a list called data. Each entry has the form:
      4 \\   [a1,a2,a3,a4,a6] (Weierstrass Coefficients)
      5 
      6 \\~2000 curves with rank 1 and conductor <1000
      7 
      8 data = [\
      9 [0,0,1,-1,0],\
     10 [0,1,1,0,0],\
     11 [1,-1,1,0,0],\
     12 [0,-1,1,-2,2],\
     13 [1,-1,0,-1,1],\
     14 [1,0,0,-2,1],\
     15 [1,0,0,-1,0],\
     16 [1,0,0,4,1],\
     17 [0,0,1,2,0],\
     18 [1,1,1,-2,0],\
     19 [1,0,1,-12,-16],\
     20 [1,0,1,-2,0],\
     21 [1,1,1,1,0],\
     22 [0,0,0,-4,4],\
     23 [1,1,1,-1,0],\
     24 [0,0,1,1,0],\
     25 [0,1,1,-117,-1245],\
     26 [0,1,1,-7,5],\
     27 [0,1,1,13,42],\
     28 [0,0,0,-1,1],\
     29 [1,-1,1,-17,30],\
     30 [1,-1,1,-2,0],\
     31 [0,1,1,-1,-1],\
     32 [1,1,0,-2,0],\
     33 [1,1,0,8,10],\
     34 [1,1,0,-7,5],\
     35 [0,1,0,-40,84],\
     36 [0,1,0,0,4],\
     37 [1,-1,1,-626,6180],\
     38 [1,-1,1,-176,-768],\
     39 [1,-1,1,-41,96],\
     40 [1,-1,1,4,6],\
     41 [1,1,0,1,1],\
     42 [0,-1,1,-887,-10143],\
     43 [0,-1,1,-7,10],\
     44 [1,0,1,2,0],\
     45 [0,1,1,-10,10],\
     46 [0,1,1,20,-890],\
     47 [0,-1,1,1,-1],\
     48 [0,1,0,-2,1],\
     49 [0,1,0,18,-11],\
     50 [0,1,0,-9,7],\
     51 [0,1,0,1,1],\
     52 [0,-1,1,-19,39],\
     53 [1,0,1,-208,-1122],\
     54 [1,0,1,-33,68],\
     55 [1,0,1,-13,156],\
     56 [1,0,1,112,-4194],\
     57 [0,-1,1,1,0],\
     58 [0,0,1,-3,4],\
     59 [0,1,0,-4,0],\
     60 [0,1,0,16,16],\
     61 [1,1,0,-31,55],\
     62 [1,1,0,-1,1],\
     63 [0,1,1,-12,2],\
     64 [0,-1,1,-1,0],\
     65 [1,1,0,-1,-1],\
     66 [1,-1,1,-12,15],\
     67 [0,-1,1,-1,-2],\
     68 [1,-1,1,-3,2],\
     69 [1,-1,1,2,6],\
     70 [0,-1,0,-5,1],\
     71 [0,1,0,-1,3],\
     72 [0,0,1,-3,2],\
     73 [0,0,1,-534,4752],\
     74 [0,0,1,6,27],\
     75 [1,-1,0,-469,4029],\
     76 [1,-1,0,-29,69],\
     77 [0,-1,1,-840,-9114],\
     78 [0,-1,1,10,6],\
     79 [0,-1,1,-1,1],\
     80 [0,-1,0,-20,-24],\
     81 [0,-1,0,-5,6],\
     82 [1,1,0,-3,1],\
     83 [1,-1,1,-9,9],\
     84 [0,1,0,-6,4],\
     85 [0,1,0,-1,15],\
     86 [1,-1,0,-6,8],\
     87 [1,-1,0,39,-19],\
     88 [0,0,1,-2,1],\
     89 [1,1,0,-6,4],\
     90 [1,0,1,-8,6],\
     91 [1,0,1,12,38],\
     92 [0,0,1,-6924,221760],\
     93 [0,0,1,-84,315],\
     94 [0,0,1,6,0],\
     95 [0,1,0,-13,15],\
     96 [0,1,0,67,79],\
     97 [0,-1,1,-148,748],\
     98 [0,-1,1,2,-2],\
     99 [0,-1,1,-3283,-74657],\
    100 [0,-1,1,-33,93],\
    101 [0,-1,1,217,-282],\
    102 [0,-1,0,-77,289],\
    103 [0,-1,0,3,1],\
    104 [0,-1,0,0,1],\
    105 [0,-1,0,-4,5],\
    106 [0,1,1,-156,700],\
    107 [0,-1,1,-5,6],\
    108 [1,0,1,-4,-3],\
    109 [1,0,1,1,-9],\
    110 [0,0,1,-3,0],\
    111 [0,0,1,-3834,-91375],\
    112 [0,0,1,-54,-88],\
    113 [0,0,1,-24,45],\
    114 [1,1,0,2,2],\
    115 [1,-1,1,-48,147],\
    116 [0,-1,0,-129,609],\
    117 [0,-1,0,-9,9],\
    118 [0,-1,0,-4,-2],\
    119 [0,-1,0,31,33],\
    120 [0,-1,0,-142,701],\
    121 [0,-1,0,-2,1],\
    122 [0,0,1,-5,4],\
    123 [1,-1,0,-3168,69430],\
    124 [1,-1,0,-198,1120],\
    125 [1,-1,0,-108,2074],\
    126 [1,-1,0,-18,4],\
    127 [0,1,0,-28,48],\
    128 [0,1,0,-3,-2],\
    129 [0,-1,1,2,0],\
    130 [1,0,0,-1,2],\
    131 [1,1,0,-794,8289],\
    132 [1,1,1,0,-2],\
    133 [1,-1,1,-232,-1286],\
    134 [1,-1,1,-27,26],\
    135 [1,-1,1,-22,44],\
    136 [1,-1,1,98,126],\
    137 [1,-1,1,-140,668],\
    138 [1,-1,1,-5,20],\
    139 [0,-1,0,-7352,245104],\
    140 [0,-1,0,-72,496],\
    141 [0,-1,0,8,-16],\
    142 [0,-1,0,-16,32],\
    143 [0,1,1,-27,55],\
    144 [0,1,1,193,-308],\
    145 [1,1,0,-373,2623],\
    146 [1,1,0,-23,33],\
    147 [1,1,0,-3,-3],\
    148 [1,1,0,7,147],\
    149 [0,-1,0,-4,8],\
    150 [1,0,1,1,0],\
    151 [1,0,1,-193,1012],\
    152 [1,0,0,-12,16],\
    153 [0,0,1,-8,-12],\
    154 [0,0,0,-12,20],\
    155 [1,0,0,-2,4],\
    156 [1,0,0,18,-104],\
    157 [0,-1,1,-6,8],\
    158 [0,1,1,-27,-85],\
    159 [0,1,1,3,2],\
    160 [1,1,0,-1297,-18530],\
    161 [1,1,0,-82,-305],\
    162 [0,1,0,-7100,-232652],\
    163 [0,1,0,-445,-3720],\
    164 [0,1,0,-100,-252],\
    165 [0,1,0,-45,100],\
    166 [0,1,0,-8,-8],\
    167 [0,1,0,2,0],\
    168 [0,0,1,-75,256],\
    169 [0,0,1,375,-12344],\
    170 [0,0,1,0,-34],\
    171 [0,0,1,0,1],\
    172 [1,0,0,-45,-119],\
    173 [1,0,0,-5,1],\
    174 [0,-1,0,3,9],\
    175 [1,0,0,-2,-1],\
    176 [0,-1,0,8,-4],\
    177 [1,-1,0,-63,209],\
    178 [1,-1,0,-3,5],\
    179 [1,1,1,-5,0],\
    180 [0,-1,0,-1,2],\
    181 [1,-1,0,-8,6],\
    182 [1,-1,0,2,0],\
    183 [1,0,0,-700,7056],\
    184 [1,0,0,-60,16],\
    185 [0,-1,0,-216,1296],\
    186 [0,-1,0,-136,-560],\
    187 [0,-1,0,-16,16],\
    188 [0,-1,0,4,0],\
    189 [1,0,0,-52,144],\
    190 [1,0,0,3,1],\
    191 [0,0,1,0,-1],\
    192 [0,0,1,0,20],\
    193 [0,0,0,1,6],\
    194 [0,0,1,-7,12],\
    195 [0,-1,1,-6435,210006],\
    196 [0,-1,1,-65,-204],\
    197 [0,-1,1,425,433],\
    198 [1,1,0,-66,180],\
    199 [1,1,0,-26,444],\
    200 [0,1,0,0,1],\
    201 [0,0,0,1,-1],\
    202 [1,1,1,-55,134],\
    203 [1,1,0,2,1],\
    204 [0,0,0,-327,2270],\
    205 [0,0,0,-12,65],\
    206 [1,-1,0,-5,-3],\
    207 [1,0,0,-24432,-1471934],\
    208 [1,0,0,-302,-2036],\
    209 [1,0,0,-22,36],\
    210 [0,1,0,-13,-21],\
    211 [0,1,0,-3,1],\
    212 [0,0,0,-2,0],\
    213 [0,0,0,8,0],\
    214 [1,1,0,3,-3],\
    215 [1,0,1,-15,22],\
    216 [1,-1,0,-2,2],\
    217 [1,0,0,1,25],\
    218 [1,-1,1,-138,656],\
    219 [1,-1,1,-133,702],\
    220 [0,0,1,-2,-1],\
    221 [0,1,0,-48,-140],\
    222 [0,1,0,-8,4],\
    223 [0,0,0,-1451,21274],\
    224 [0,0,0,-91,330],\
    225 [0,0,0,-11,-6],\
    226 [0,0,0,-11,890],\
    227 [0,-1,1,-26,68],\
    228 [1,-1,0,-2846,59156],\
    229 [1,-1,0,-2,0],\
    230 [1,-1,0,8,-6],\
    231 [1,0,0,-7,9],\
    232 [1,-1,1,-1480,22272],\
    233 [1,-1,1,-730,-7228],\
    234 [1,-1,1,-105,272],\
    235 [1,-1,1,20,22],\
    236 [1,0,1,0,-1],\
    237 [1,0,0,-1,9],\
    238 [0,0,0,-412,3316],\
    239 [0,-1,0,-1,5],\
    240 [1,1,1,-255,1461],\
    241 [1,1,1,-15,21],\
    242 [1,0,0,-76,-19],\
    243 [1,0,0,19,0],\
    244 [1,1,0,-93,-378],\
    245 [1,1,0,2,-17],\
    246 [1,1,0,-33,61],\
    247 [1,1,1,13,177],\
    248 [0,0,0,-12,0],\
    249 [0,0,0,3,0],\
    250 [0,0,0,-291,-1910],\
    251 [0,0,0,-156,736],\
    252 [0,0,0,-21,-20],\
    253 [0,0,0,69,-146],\
    254 [1,-1,1,-26209,-1626560],\
    255 [1,-1,1,-1644,-24922],\
    256 [1,-1,1,-199,-68272],\
    257 [1,-1,1,-199,510],\
    258 [1,-1,0,-70,-204],\
    259 [1,-1,0,10,-700],\
    260 [1,1,1,-18,-36],\
    261 [1,1,1,-3,0],\
    262 [1,0,1,-138,592],\
    263 [1,0,1,2,32],\
    264 [0,-1,0,-9,13],\
    265 [0,-1,0,-33,85],\
    266 [0,0,1,-81,290],\
    267 [1,-1,1,1,0],\
    268 [1,-1,0,12,-19],\
    269 [1,-1,0,1,-1],\
    270 [1,0,0,-19,33],\
    271 [0,-1,0,-13,22],\
    272 [0,-1,0,12,72],\
    273 [1,-1,1,0,3],\
    274 [1,1,1,-230,1251],\
    275 [1,1,1,1650,-27389],\
    276 [0,1,1,-6,2],\
    277 [0,1,1,-197,-208],\
    278 [0,1,0,-21,31],\
    279 [0,-1,0,-8,16],\
    280 [0,1,0,-1120,15604],\
    281 [0,1,0,0,-76],\
    282 [1,-1,0,-1017,8883],\
    283 [1,-1,0,-927,11097],\
    284 [1,-1,0,-387,-2835],\
    285 [1,-1,0,-27,-27],\
    286 [0,-1,0,-21,49],\
    287 [1,0,0,-6,9],\
    288 [1,0,0,-2046,15376],\
    289 [1,0,0,-1706,26980],\
    290 [1,0,0,-106,420],\
    291 [1,0,0,454,1876],\
    292 [0,-1,0,-3,0],\
    293 [0,-1,0,12,-12],\
    294 [0,1,0,-60,144],\
    295 [0,1,0,5,14],\
    296 [1,-1,0,13,-11],\
    297 [1,-1,1,-1013,12656],\
    298 [1,-1,1,-68,182],\
    299 [1,-1,1,-23,-34],\
    300 [1,-1,1,157,992],\
    301 [0,0,0,-7,-2],\
    302 [1,1,0,7,-9],\
    303 [1,1,1,-12,45],\
    304 [0,1,0,-165,763],\
    305 [0,1,0,-145,975],\
    306 [0,1,0,-5,-5],\
    307 [0,1,0,15,-17],\
    308 [0,0,0,-428,-3408],\
    309 [0,0,0,-28,-48],\
    310 [0,0,0,-8,8],\
    311 [0,0,0,52,-272],\
    312 [1,-1,0,-238,1470],\
    313 [1,-1,0,-8,44],\
    314 [1,0,0,-174,868],\
    315 [1,0,0,-14,4],\
    316 [0,0,0,-189,-999],\
    317 [0,0,0,-9,9],\
    318 [0,-1,1,-53,-132],\
    319 [0,-1,1,-3,3],\
    320 [0,1,1,-1333,-19131],\
    321 [0,1,1,-83,244],\
    322 [1,-1,0,-80,-256],\
    323 [1,0,0,-6,4],\
    324 [1,0,0,4,-3],\
    325 [0,0,0,-11,-10],\
    326 [0,0,0,29,-66],\
    327 [1,1,0,-1602,24024],\
    328 [1,1,0,-102,324],\
    329 [1,1,0,-22,-44],\
    330 [1,1,0,118,1776],\
    331 [1,0,0,-5,4],\
    332 [1,-1,1,-14,-14],\
    333 [1,-1,1,1,-2],\
    334 [0,0,1,-16860,842625],\
    335 [0,0,1,-210,1134],\
    336 [0,0,1,-30,-63],\
    337 [1,-1,0,-123,494],\
    338 [1,-1,0,12,35],\
    339 [0,0,1,-2,2],\
    340 [0,-1,0,-12544,544960],\
    341 [0,-1,0,-784,8704],\
    342 [0,-1,0,-624,-5760],\
    343 [0,-1,0,-544,13888],\
    344 [0,-1,0,-64,64],\
    345 [0,-1,0,16,0],\
    346 [1,-1,0,-35944,-2868878],\
    347 [1,-1,0,-454,5812],\
    348 [1,-1,0,-389,-2859],\
    349 [1,-1,0,1,1],\
    350 [1,1,1,-322,2127],\
    351 [1,1,1,3,-5],\
    352 [0,1,1,-2,2],\
    353 [0,1,1,-441,3422],\
    354 [0,0,0,-28,57],\
    355 [0,0,0,-23,78],\
    356 [1,-1,0,-33,-65],\
    357 [1,-1,0,-3,1],\
    358 [1,-1,0,-3852,92988],\
    359 [1,-1,0,-3762,97470],\
    360 [1,-1,0,-72,0],\
    361 [1,-1,0,288,-216],\
    362 [0,0,0,4,4],\
    363 [0,1,1,-100,406],\
    364 [0,1,1,-1,1],\
    365 [1,1,1,-7,-3],\
    366 [0,1,1,2,0],\
    367 [0,-1,0,2,1],\
    368 [0,1,0,-50,129],\
    369 [1,1,0,-45,-185],\
    370 [1,1,0,5,5],\
    371 [1,-1,1,-180,1047],\
    372 [0,0,0,8,112],\
    373 [0,-1,0,3,-11],\
    374 [0,-1,0,-45,133],\
    375 [0,1,0,3,11],\
    376 [1,1,0,-715,7069],\
    377 [1,1,1,-5,11],\
    378 [0,-1,0,4,-8],\
    379 [0,1,1,-42,110],\
    380 [0,-1,1,-5,-16],\
    381 [1,-1,1,-7,8],\
    382 [1,0,1,-23,39],\
    383 [0,0,0,-963,11502],\
    384 [0,0,0,-63,162],\
    385 [0,0,0,-18,-27],\
    386 [0,0,0,117,918],\
    387 [0,0,1,-13718,-619025],\
    388 [0,0,1,-38,90],\
    389 [1,1,0,-4,2],\
    390 [1,1,1,6,7],\
    391 [0,1,0,-5,7],\
    392 [0,0,0,-584,5444],\
    393 [1,0,1,-5,20],\
    394 [1,0,1,40,-538],\
    395 [1,1,1,-32,65],\
    396 [0,0,0,-55,157],\
    397 [0,-1,0,-18,43],\
    398 [0,-1,0,2,-1],\
    399 [0,0,0,-35,-62],\
    400 [0,0,0,5,-6],\
    401 [0,1,0,0,-1],\
    402 [0,0,1,6,13],\
    403 [1,-1,0,-395,-2925],\
    404 [1,-1,0,-25,-39],\
    405 [1,-1,0,-5,5],\
    406 [1,-1,0,25,-209],\
    407 [1,1,0,-35,-98],\
    408 [0,-1,0,-6,9],\
    409 [0,1,0,-2,9],\
    410 [0,1,1,-2,-2],\
    411 [1,-1,0,-32,0],\
    412 [1,-1,0,128,-96],\
    413 [1,-1,0,-158,-725],\
    414 [1,-1,0,-13,0],\
    415 [1,-1,0,-8,11],\
    416 [1,-1,0,52,-39],\
    417 [1,-1,0,-141,681],\
    418 [1,-1,0,-96,1088],\
    419 [1,-1,0,849,-25939],\
    420 [1,-1,0,0,4],\
    421 [0,0,0,-103,-402],\
    422 [0,0,0,-8,-3],\
    423 [0,1,1,-11,-16],\
    424 [0,-1,0,-3,3],\
    425 [0,-1,0,7,9],\
    426 [0,1,0,-35,69],\
    427 [0,1,0,-25,119],\
    428 [1,0,0,-55,150],\
    429 [1,0,0,0,7],\
    430 [1,-1,1,-10267,402966],\
    431 [1,-1,1,-697,5294],\
    432 [1,-1,1,-642,6416],\
    433 [1,-1,1,-37,124],\
    434 [1,-1,1,-2,2],\
    435 [1,-1,0,-15,-46],\
    436 [1,1,0,-483,-4293],\
    437 [1,1,0,-33,-63],\
    438 [1,1,0,-13,13],\
    439 [1,1,0,97,-297],\
    440 [0,0,0,-7,7],\
    441 [0,-1,0,-16,29],\
    442 [0,0,0,-14651,-682570],\
    443 [0,0,0,-2891,47334],\
    444 [0,0,0,-931,-10290],\
    445 [0,0,0,49,-686],\
    446 [0,0,0,-111,214],\
    447 [0,0,0,24,25],\
    448 [1,1,1,-48,90],\
    449 [1,1,1,-13,-22],\
    450 [1,1,0,-1925,31458],\
    451 [1,1,0,-210,-441],\
    452 [0,0,0,5,10],\
    453 [0,-1,0,-2008,35312],\
    454 [0,-1,0,-1208,-19088],\
    455 [0,-1,0,-8,112],\
    456 [0,-1,0,8792,140912],\
    457 [0,0,0,-2675,53250],\
    458 [0,0,0,-175,750],\
    459 [0,0,0,-50,-125],\
    460 [0,0,0,325,4250],\
    461 [1,1,0,-2,-12],\
    462 [1,0,1,-10255,-438718],\
    463 [1,0,1,-145,692],\
    464 [1,0,1,800,1070],\
    465 [0,0,0,-8,4],\
    466 [0,0,1,-3,-2],\
    467 [1,-1,1,-2027,35776],\
    468 [1,-1,1,-2,-26],\
    469 [0,0,1,-108,-412],\
    470 [0,0,1,-18,29],\
    471 [1,-1,0,-225,-1250],\
    472 [1,-1,0,0,1],\
    473 [1,-1,0,-4942,134964],\
    474 [1,-1,0,-302,2260],\
    475 [1,0,1,-15,210],\
    476 [1,0,1,130,-5648],\
    477 [1,1,1,-102,355],\
    478 [0,1,0,-17,51],\
    479 [1,-1,0,-214,1260],\
    480 [1,-1,0,-14,20],\
    481 [1,0,0,-16,0],\
    482 [1,0,0,64,16],\
    483 [1,-1,0,-2223,-39785],\
    484 [1,-1,0,-963,11371],\
    485 [1,-1,0,-153,-455],\
    486 [1,-1,0,27,-59],\
    487 [1,-1,1,-1532,23455],\
    488 [1,-1,1,-92,415],\
    489 [0,-1,0,0,4],\
    490 [1,1,1,66,-5],\
    491 [1,-1,0,1,-3],\
    492 [0,0,1,-9,10],\
    493 [0,0,1,-237,1404],\
    494 [0,0,1,-12,4],\
    495 [1,-1,0,-6768,-212625],\
    496 [1,-1,0,-558,-891],\
    497 [1,-1,0,-423,-3240],\
    498 [1,-1,0,-18,-81],\
    499 [1,0,0,-213,-1208],\
    500 [1,0,0,-88,-2583],\
    501 [1,0,0,-3,2],\
    502 [1,1,0,-75,250],\
    503 [1,-1,0,-2267,-40984],\
    504 [1,-1,0,-142,-609],\
    505 [1,-1,0,-17,-1734],\
    506 [1,-1,0,-17,16],\
    507 [1,1,0,-4606,118420],\
    508 [1,1,0,-286,1780],\
    509 [1,0,0,-28,-59],\
    510 [1,0,1,-8,7],\
    511 [0,-1,0,3,-2],\
    512 [1,1,1,-13,8],\
    513 [1,1,1,2,2],\
    514 [1,0,0,-6864,218313],\
    515 [1,0,0,-3009,-61770],\
    516 [1,0,0,-474,2619],\
    517 [1,0,0,-429,3384],\
    518 [1,0,0,-24,63],\
    519 [1,0,0,1341,18228],\
    520 [1,-1,0,-20,40],\
    521 [1,-1,0,16,-10],\
    522 [1,0,0,-5626,-162894],\
    523 [1,0,0,-36,-440],\
    524 [1,0,0,4,16],\
    525 [1,0,0,-1415,20617],\
    526 [1,0,0,0,-1],\
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   2005 [1,0,1,-1611,-39690],\
   2006 [1,0,1,164,922],\
   2007 [1,0,0,-52238,-4587140],\
   2008 [1,0,0,-52198,-4594524],\
   2009 [1,0,0,-3238,65508],\
   2010 [1,0,0,-678,-5660],\
   2011 [1,-1,1,-1516,23091],\
   2012 [1,-1,1,-96,371],\
   2013 [1,-1,1,-16,-13],\
   2014 [1,-1,1,44,1267],\
   2015 [0,1,1,-15,19],\
   2016 [0,1,1,85,64],\
   2017 [0,1,0,-12,36],\
   2018 [0,1,0,108,-876],\
   2019 [0,1,0,164,-8764],\
   2020 [0,-1,1,-18,36],\
   2021 [1,-1,1,-8,10],\
   2022 [1,-1,0,-69,-208]]