Abstract
We present the results of our search for the orders of Tate–Shafarevich groups for the quadratic twists of \(E=X_0(49)\).
Dedicated to John Coates on his seventieth birthday.
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References
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The PARI Group, PARI/GP version 2.7.2, Bordeaux (2014). http://pari.math.u-bordeaux.fr/
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Appendices
Appendix: The Algorithm and the Implementation
The strategy is to use the construction described in the end of Sect. 2 to compute the coefficients \(a_d\) for d satisfying condition (**) up to \(32 \cdot 10^{9}\), and use Corollary 1.
We present our algorithm using computer algebra system PARI/GP [18].
The key point of the above implementation is using the qfrep function. Recall that qfrep ( q , B ) returns the vector whose i-th entry \((1 \leqslant i \leqslant B)\) is half the number of vectors v such that \(q(v) = i\). Routine qfrep relies on qfminim function which enumerates, using the Fincke–Pohst algorithm, the vectors v for which \(q(v) \leqslant B\).
We used the above PARI/GP script (with small modifications) to compute sha( B ) for \(B=5\cdot 10^7\) (on standard desktop PC). We made further progress implementing qfrep function in C++ language. Our routine directly enumerates all vectors v for which \(f_i(v)\leqslant B\) for \(i=1,2,3,4\). Such a straightforward approach is more effective for forms \(f_1, f_2, f_3, f_4\) than the sophisticated Fincke–Pohst algorithm used by PARI/GP. Moreover, we made some optimizations. Most important among them are:
-
time optimization—exploiting symmetries of \(f_1, f_2, f_3, f_4\),
-
memory optimization—storing in memory only values of \(a_d\) for \(d\equiv 1\pmod 4\).
Enumerating numbers satisfying condition (**) (and factoring them) also takes some time. One can be speed it up by filtering out numbers which are not square-free. We did it by using a modified sieve of Eratosthenes. However, the real bottleneck is computing qfrep.
Let us add that our algorithm is quite easily parallelizable. It appears that performing computation in parallel will make possible to increase B substantially compared to our achievement.
Tables
For each positive integer \(k\leqslant 1793\), the column headed \(d_1\) gives the smallest integer d for which . One interesting observation is that all odd orders are realized by the integers \(d_1\) satisfying the condition (*).
k | \(d_1\) | k | \(d_1\) | k | \(d_1\) | k | \(d_1\) |
---|---|---|---|---|---|---|---|
1 | 1 | 2 | 93 | 3 | 73 | 4 | 177 |
5 | 257 | 6 | 933 | 7 | 929 | 8 | 4337 |
9 | 2281 | 10 | 6073 | 11 | 3169 | 12 | 6609 |
13 | 5897 | 14 | 14177 | 15 | 12241 | 16 | 20497 |
17 | 10937 | 18 | 19713 | 19 | 5641 | 20 | 52257 |
21 | 18793 | 22 | 40769 | 23 | 31513 | 24 | 63473 |
25 | 26249 | 26 | 55617 | 27 | 23369 | 28 | 63849 |
29 | 62929 | 30 | 121881 | 31 | 49993 | 32 | 152769 |
33 | 65609 | 34 | 100857 | 35 | 62401 | 36 | 167073 |
37 | 98257 | 38 | 322921 | 39 | 96353 | 40 | 226913 |
41 | 133769 | 42 | 206273 | 43 | 151273 | 44 | 734001 |
45 | 110977 | 46 | 337681 | 47 | 129457 | 48 | 498129 |
49 | 253553 | 50 | 549817 | 51 | 152953 | 52 | 518137 |
53 | 152249 | 54 | 702353 | 55 | 291457 | 56 | 612529 |
57 | 247369 | 58 | 673817 | 59 | 368857 | 60 | 953313 |
61 | 365249 | 62 | 964793 | 63 | 626377 | 64 | 847793 |
65 | 290657 | 66 | 1319649 | 67 | 527729 | 68 | 1217049 |
69 | 536017 | 70 | 1091841 | 71 | 957361 | 72 | 2060353 |
73 | 637297 | 74 | 1501329 | 75 | 423097 | 76 | 1135649 |
77 | 1465469 | 78 | 1707729 | 79 | 955769 | 80 | 1827193 |
81 | 570113 | 82 | 2874369 | 83 | 682009 | 84 | 1234137 |
85 | 1101593 | 86 | 2827553 | 87 | 1899481 | 88 | 2229529 |
89 | 1885673 | 90 | 2341817 | 91 | 1323689 | 92 | 2799217 |
93 | 1381337 | 94 | 3018513 | 95 | 1242169 | 96 | 2904801 |
97 | 1917697 | 98 | 4294313 | 99 | 1790897 | 100 | 3567881 |
101 | 1625321 | 102 | 4518273 | 103 | 1866857 | 104 | 5884041 |
105 | 1781569 | 106 | 4184049 | 107 | 2915713 | 108 | 6165329 |
109 | 2182249 | 110 | 4479897 | 111 | 3647689 | 112 | 4909017 |
113 | 1465313 | 114 | 5427489 | 115 | 2761841 | 116 | 6469849 |
117 | 2687257 | 118 | 6350073 | 119 | 3393449 | 120 | 4884177 |
121 | 3524041 | 122 | 7486329 | 123 | 3485513 | 124 | 7240809 |
125 | 3613193 | 126 | 4935001 | 127 | 4229657 | 128 | 7353921 |
129 | 3486257 | 130 | 7753601 | 131 | 4459601 | 132 | 4110177 |
133 | 4693177 | 134 | 8832657 | 135 | 3247313 | 136 | 8314777 |
137 | 4296977 | 138 | 10538889 | 139 | 5507297 | 140 | 11180073 |
141 | 3688081 | 142 | 10114889 | 143 | 6025801 | 144 | 6302409 |
145 | 3653369 | 146 | 14245449 | 147 | 5294833 | 148 | 11250761 |
149 | 3106921 | 150 | 10362081 | 151 | 5946337 | 152 | 13688313 |
153 | 6790073 | 154 | 10240521 | 155 | 7491361 | 156 | 15464089 |
157 | 4103641 | 158 | 10692817 | 159 | 7016777 | 160 | 11928953 |
161 | 6718193 | 162 | 15799897 | 163 | 6645721 | 164 | 13927593 |
165 | 10108297 | 166 | 12971121 | 167 | 6120929 | 168 | 19275657 |
169 | 10688753 | 170 | 21326937 | 171 | 6078337 | 172 | 9310449 |
173 | 8860361 | 174 | 16962969 | 175 | 9539281 | 176 | 23387809 |
177 | 9032609 | 178 | 18940849 | 179 | 9425113 | 180 | 24630321 |
181 | 9843529 | 182 | 21259921 | 183 | 5634809 | 184 | 20181561 |
185 | 15130393 | 186 | 18589729 | 187 | 10534921 | 188 | 26802777 |
k | \(d_1\) | k | \(d_1\) | k | \(d_1\) | k | \(d_1\) |
---|---|---|---|---|---|---|---|
189 | 9128969 | 190 | 18017673 | 191 | 15028201 | 192 | 27317361 |
193 | 17238961 | 194 | 27932529 | 195 | 13267873 | 196 | 18475521 |
197 | 19233889 | 198 | 32768481 | 199 | 18412817 | 200 | 17722329 |
201 | 14262433 | 202 | 35158337 | 203 | 17371513 | 204 | 31450409 |
205 | 15861473 | 206 | 55430673 | 207 | 17282137 | 208 | 25378833 |
209 | 16847953 | 210 | 20905977 | 211 | 13179577 | 212 | 30805273 |
213 | 17764913 | 214 | 47121369 | 215 | 14565193 | 216 | 24423177 |
217 | 17119073 | 218 | 31973313 | 219 | 20660377 | 220 | 30225721 |
221 | 20663593 | 222 | 32356041 | 223 | 24736529 | 224 | 31522353 |
225 | 17100961 | 226 | 44057561 | 227 | 23429761 | 228 | 61238433 |
229 | 20035217 | 230 | 30310809 | 231 | 17521937 | 232 | 45713721 |
233 | 26153209 | 234 | 53720529 | 235 | 19521001 | 236 | 30965713 |
237 | 17479313 | 238 | 45580921 | 239 | 19624729 | 240 | 59076249 |
241 | 24796313 | 242 | 46666337 | 243 | 15196457 | 244 | 47964921 |
245 | 24126161 | 246 | 67880649 | 247 | 15737417 | 248 | 59498961 |
249 | 27527393 | 250 | 67310681 | 251 | 31900529 | 252 | 71401089 |
253 | 21488809 | 254 | 42480201 | 255 | 21141041 | 256 | 62559121 |
257 | 21436001 | 258 | 44968137 | 259 | 23661529 | 260 | 68143553 |
261 | 28188257 | 262 | 83482809 | 263 | 45616297 | 264 | 74407953 |
265 | 33502577 | 266 | 76802441 | 267 | 45721681 | 268 | 56817777 |
269 | 30511001 | 270 | 80564961 | 271 | 42257857 | 272 | 65262849 |
273 | 30407369 | 274 | 51619593 | 275 | 34562401 | 276 | 86165913 |
277 | 28530241 | 278 | 100543353 | 279 | 29771201 | 280 | 55275609 |
281 | 33775801 | 282 | 102490809 | 283 | 38382041 | 284 | 89933937 |
285 | 27594521 | 286 | 94586473 | 287 | 41793233 | 288 | 84939537 |
289 | 47313209 | 290 | 94270929 | 291 | 25854097 | 292 | 74695377 |
293 | 49080337 | 294 | 129084873 | 295 | 48796537 | 296 | 97992497 |
297 | 41571113 | 298 | 108653521 | 299 | 63138337 | 300 | 114844137 |
301 | 28987073 | 302 | 117318657 | 303 | 70938377 | 304 | 120142353 |
305 | 54726241 | 306 | 106517777 | 307 | 62983121 | 308 | 108212241 |
309 | 54211177 | 310 | 132195057 | 311 | 57847201 | 312 | 107424529 |
313 | 64804081 | 314 | 136571681 | 315 | 67153729 | 316 | 72567049 |
317 | 39044641 | 318 | 103714801 | 319 | 68778097 | 320 | 124306361 |
321 | 67515881 | 322 | 99630969 | 323 | 42683441 | 324 | 126667249 |
325 | 81725521 | 326 | 128132673 | 327 | 22476089 | 328 | 69421713 |
329 | 82804153 | 330 | 131429937 | 331 | 35634569 | 332 | 103211529 |
333 | 53915137 | 334 | 127330809 | 335 | 58071737 | 336 | 145127361 |
337 | 82549193 | 338 | 127179537 | 339 | 81256817 | 340 | 212734713 |
341 | 94701017 | 342 | 120707049 | 343 | 74528177 | 344 | 180657537 |
345 | 82682417 | 346 | 163723673 | 347 | 83092201 | 348 | 127062681 |
349 | 64266313 | 350 | 158078897 | 351 | 45363041 | 352 | 165215121 |
353 | 86228729 | 354 | 150696393 | 355 | 74605177 | 356 | 158006489 |
357 | 96923641 | 358 | 168255201 | 359 | 101236001 | 360 | 199147433 |
361 | 61967953 | 362 | 208401153 | 363 | 75284753 | 364 | 179223529 |
365 | 81361481 | 366 | 190068321 | 367 | 128132273 | 368 | 233660249 |
369 | 104815049 | 370 | 166966409 | 371 | 78813817 | 372 | 166934289 |
373 | 85236353 | 374 | 259338137 | 375 | 100124561 | 376 | 150853497 |
377 | 86855849 | 378 | 195302193 | 379 | 117023129 | 380 | 170908593 |
k | \(d_1\) | k | \(d_1\) | k | \(d_1\) | k | \(d_1\) |
---|---|---|---|---|---|---|---|
381 | 119282593 | 382 | 229290681 | 383 | 124208657 | 384 | 250172553 |
385 | 84921329 | 386 | 242076657 | 387 | 81585241 | 388 | 198355873 |
389 | 111613529 | 390 | 317098497 | 391 | 68312473 | 392 | 164772969 |
393 | 137121073 | 394 | 317478617 | 395 | 184317257 | 396 | 267280841 |
397 | 157495033 | 398 | 198455217 | 399 | 95602057 | 400 | 201599049 |
401 | 141095993 | 402 | 315033657 | 403 | 162887129 | 404 | 252130649 |
405 | 129060721 | 406 | 239496657 | 407 | 75453481 | 408 | 309730961 |
409 | 145253513 | 410 | 263668161 | 411 | 114535441 | 412 | 258161649 |
413 | 70924577 | 414 | 201012681 | 415 | 156669217 | 416 | 348700969 |
417 | 133510961 | 418 | 255577281 | 419 | 133337329 | 420 | 243488697 |
421 | 163103713 | 422 | 349629081 | 423 | 129344561 | 424 | 248961057 |
425 | 113424457 | 426 | 158340513 | 427 | 161716729 | 428 | 457284881 |
429 | 149221609 | 430 | 306979737 | 431 | 188287097 | 432 | 263466921 |
433 | 113174249 | 434 | 379096881 | 435 | 116677553 | 436 | 341102721 |
437 | 138979921 | 438 | 344625297 | 439 | 142217729 | 440 | 402660633 |
441 | 173153249 | 442 | 337717857 | 443 | 124106569 | 444 | 303693153 |
445 | 187920529 | 446 | 348974817 | 447 | 141120313 | 448 | 374142729 |
449 | 169920161 | 450 | 415195257 | 451 | 119896241 | 452 | 456636161 |
453 | 172998929 | 454 | 370842873 | 455 | 186067649 | 456 | 284126217 |
457 | 211471489 | 458 | 322781001 | 459 | 175500121 | 460 | 424009721 |
461 | 175321193 | 462 | 383282377 | 463 | 227914553 | 464 | 411552969 |
465 | 151632193 | 466 | 405886881 | 467 | 160286201 | 468 | 493254001 |
469 | 133377289 | 470 | 381535529 | 471 | 150114793 | 472 | 467868809 |
473 | 199481561 | 474 | 367048641 | 475 | 231417217 | 476 | 428448673 |
477 | 216300353 | 478 | 464970993 | 479 | 182712193 | 480 | 377366529 |
481 | 203950673 | 482 | 481370849 | 483 | 208906417 | 484 | 466546089 |
485 | 221357009 | 486 | 611992833 | 487 | 213593561 | 488 | 445591689 |
489 | 152807297 | 490 | 365071281 | 491 | 227448577 | 492 | 324398649 |
493 | 152292209 | 494 | 327557561 | 495 | 158411977 | 496 | 455620881 |
497 | 235272161 | 498 | 298850577 | 499 | 159619337 | 500 | 487175313 |
501 | 87873329 | 502 | 433020569 | 503 | 209599633 | 504 | 468120129 |
505 | 233102873 | 506 | 680179593 | 507 | 248435521 | 508 | 409342217 |
509 | 167807489 | 510 | 499217601 | 511 | 201167833 | 512 | 502073881 |
513 | 256670657 | 514 | 601398561 | 515 | 265990297 | 516 | 591746313 |
517 | 225790673 | 518 | 474463697 | 519 | 244563961 | 520 | 560288513 |
521 | 191110121 | 522 | 724247857 | 523 | 259344209 | 524 | 455548713 |
525 | 263635321 | 526 | 426422649 | 527 | 313004473 | 528 | 484397841 |
529 | 231144521 | 530 | 556803769 | 531 | 257557873 | 532 | 652226129 |
533 | 328620697 | 534 | 579162081 | 535 | 335578081 | 536 | 618274897 |
537 | 312412721 | 538 | 888968217 | 539 | 291056657 | 540 | 577412049 |
541 | 243334657 | 542 | 871280337 | 543 | 389624233 | 544 | 484212369 |
545 | 320529089 | 546 | 874368489 | 547 | 260262113 | 548 | 784207257 |
549 | 298280401 | 550 | 598627929 | 551 | 359759921 | 552 | 374213393 |
553 | 217628393 | 554 | 840530793 | 555 | 258609433 | 556 | 789059793 |
557 | 376545137 | 558 | 899077953 | 559 | 310206713 | 560 | 849373977 |
561 | 276498809 | 562 | 720256769 | 563 | 359961713 | 564 | 605796249 |
565 | 505198489 | 566 | 660711921 | 567 | 374576513 | 568 | 693060153 |
569 | 285955057 | 570 | 582042129 | 571 | 194086553 | 572 | 971076633 |
k | \(d_1\) | k | \(d_1\) | k | \(d_1\) | k | \(d_1\) |
---|---|---|---|---|---|---|---|
573 | 269512417 | 574 | 812238113 | 575 | 394565449 | 576 | 531793497 |
577 | 376637449 | 578 | 558769737 | 579 | 177118241 | 580 | 800651713 |
581 | 314235617 | 582 | 971046561 | 583 | 278549137 | 584 | 713190633 |
585 | 343632217 | 586 | 910562577 | 587 | 299414377 | 588 | 806580569 |
589 | 339441217 | 590 | 969882153 | 591 | 404910817 | 592 | 1180596561 |
593 | 445593769 | 594 | 746080449 | 595 | 366680009 | 596 | 588145881 |
597 | 499709489 | 598 | 1000317921 | 599 | 453223081 | 600 | 857027697 |
601 | 482163793 | 602 | 709173529 | 603 | 464499577 | 604 | 1085969329 |
605 | 583993777 | 606 | 611140737 | 607 | 477113129 | 608 | 767941113 |
609 | 487518937 | 610 | 1205784057 | 611 | 345190481 | 612 | 1042100481 |
613 | 433569953 | 614 | 759143361 | 615 | 295290449 | 616 | 1060039817 |
617 | 271819777 | 618 | 916876041 | 619 | 399513761 | 620 | 833937729 |
621 | 351775609 | 622 | 1014335921 | 623 | 461803457 | 624 | 1063477353 |
625 | 604768433 | 626 | 1195237257 | 627 | 355313929 | 628 | 1130731233 |
629 | 356217217 | 630 | 1088179233 | 631 | 523841737 | 632 | 1296431537 |
633 | 373251833 | 634 | 1046573089 | 635 | 533158321 | 636 | 1129661633 |
637 | 437185369 | 638 | 1001125833 | 639 | 690863353 | 640 | 1049542113 |
641 | 393691721 | 642 | 1219662481 | 643 | 442113409 | 644 | 866810121 |
645 | 473179657 | 646 | 918013177 | 647 | 422803841 | 648 | 925311633 |
649 | 628298753 | 650 | 908022641 | 651 | 416028409 | 652 | 668155161 |
653 | 639545801 | 654 | 788481633 | 655 | 456778633 | 656 | 1253196393 |
657 | 553788233 | 658 | 1011782193 | 659 | 559500833 | 660 | 1095831129 |
661 | 502398097 | 662 | 1556083761 | 663 | 540229913 | 664 | 1511104233 |
665 | 487247689 | 666 | 1530974993 | 667 | 419408113 | 668 | 1006678033 |
669 | 557895281 | 670 | 1502333529 | 671 | 748804241 | 672 | 954448401 |
673 | 431472649 | 674 | 1503531033 | 675 | 567004513 | 676 | 1390863849 |
677 | 486471529 | 678 | 845750049 | 679 | 563530553 | 680 | 1401556137 |
681 | 472466377 | 682 | 1131586201 | 683 | 774729497 | 684 | 1153585497 |
685 | 424046153 | 686 | 1292386929 | 687 | 384060241 | 688 | 969826881 |
689 | 677852257 | 690 | 1146387201 | 691 | 472867673 | 692 | 1284311337 |
693 | 590461073 | 694 | 1448799433 | 695 | 552263017 | 696 | 1430856177 |
697 | 487192513 | 698 | 1065709793 | 699 | 789946601 | 700 | 1272089361 |
701 | 617717609 | 702 | 1305583953 | 703 | 622046417 | 704 | 1815425929 |
705 | 510979577 | 706 | 886583769 | 707 | 571233809 | 708 | 1259122593 |
709 | 634656713 | 710 | 1451816193 | 711 | 769553489 | 712 | 1902558001 |
713 | 919520761 | 714 | 1671253809 | 715 | 555591601 | 716 | 1737287913 |
717 | 690268577 | 718 | 1691545857 | 719 | 758546561 | 720 | 1417870169 |
721 | 626618809 | 722 | 2027470713 | 723 | 645226937 | 724 | 1660673337 |
725 | 357461201 | 726 | 1159545633 | 727 | 803008321 | 728 | 1172959881 |
729 | 645947833 | 730 | 1532369217 | 731 | 786522889 | 732 | 1220017089 |
733 | 822906313 | 734 | 1812159633 | 735 | 723749137 | 736 | 1677838089 |
737 | 797421113 | 738 | 1334772609 | 739 | 533625049 | 740 | 2156546177 |
741 | 898149521 | 742 | 1908161193 | 743 | 777702721 | 744 | 1475258537 |
745 | 764284097 | 746 | 1932838161 | 747 | 774980849 | 748 | 1597045321 |
749 | 805117801 | 750 | 1649813817 | 751 | 912356273 | 752 | 1922828889 |
753 | 535552681 | 754 | 2049963393 | 755 | 618088673 | 756 | 1439459169 |
757 | 1012636441 | 758 | 1734170937 | 759 | 704129537 | 760 | 2013014697 |
761 | 673372577 | 762 | 2328013713 | 763 | 832572401 | 764 | 1695155961 |
k | \(d_1\) | k | \(d_1\) | k | \(d_1\) | k | \(d_1\) |
---|---|---|---|---|---|---|---|
765 | 920887169 | 766 | 1176556089 | 767 | 831201097 | 768 | 1843728657 |
769 | 786715537 | 770 | 2185812897 | 771 | 637134697 | 772 | 1860573801 |
773 | 623324777 | 774 | 2428781793 | 775 | 854043041 | 776 | 1309574841 |
777 | 413798201 | 778 | 2198450769 | 779 | 790908361 | 780 | 1722152721 |
781 | 1058446409 | 782 | 1562389249 | 783 | 673866737 | 784 | 1600255897 |
785 | 859210817 | 786 | 2201361121 | 787 | 845971769 | 788 | 2347753809 |
789 | 1009504417 | 790 | 2403252273 | 791 | 780662161 | 792 | 2485110489 |
793 | 991564633 | 794 | 1670427609 | 795 | 988737601 | 796 | 1304526953 |
797 | 851169289 | 798 | 2039945329 | 799 | 909129017 | 800 | 1739103809 |
801 | 936222409 | 802 | 1575219297 | 803 | 1063972649 | 804 | 2389858689 |
805 | 810098873 | 806 | 2374269969 | 807 | 838474873 | 808 | 1497563313 |
809 | 980245633 | 810 | 1902373617 | 811 | 1228354273 | 812 | 1950931833 |
813 | 636479009 | 814 | 1263787449 | 815 | 1314825769 | 816 | 2641549089 |
817 | 691199521 | 818 | 2555306121 | 819 | 839732129 | 820 | 2539277841 |
821 | 880782593 | 822 | 2059914081 | 823 | 1530012697 | 824 | 1619827449 |
825 | 1079910089 | 826 | 2651407089 | 827 | 877717273 | 828 | 2528252409 |
829 | 1123787593 | 830 | 2115598521 | 831 | 879991633 | 832 | 2757188417 |
833 | 910187017 | 834 | 2576804673 | 835 | 1162667929 | 836 | 2967468729 |
837 | 1194444817 | 838 | 1496186553 | 839 | 1092092489 | 840 | 2828807529 |
841 | 1168598161 | 842 | 1741340841 | 843 | 1217448433 | 844 | 2961741017 |
845 | 1055512121 | 846 | 2086860777 | 847 | 1176846961 | 848 | 2857144161 |
849 | 1145287009 | 850 | 3129250569 | 851 | 1359540241 | 852 | 2734116361 |
853 | 1270901369 | 854 | 2849667881 | 855 | 1316031401 | 856 | 4109398617 |
857 | 1192513873 | 858 | 2589301401 | 859 | 1747673497 | 860 | 2374378953 |
861 | 1665814057 | 862 | 2457866009 | 863 | 864297857 | 864 | 2597941641 |
865 | 1105681657 | 866 | 2126163873 | 867 | 1474091393 | 868 | 2422942097 |
869 | 813330241 | 870 | 2587316257 | 871 | 1215280433 | 872 | 2864765121 |
873 | 1055546617 | 874 | 2955604521 | 875 | 1588324849 | 876 | 2540162769 |
877 | 1166140897 | 878 | 1740197649 | 879 | 1071471217 | 880 | 3251124937 |
881 | 1121666929 | 882 | 2896790241 | 883 | 1071890161 | 884 | 2631081489 |
885 | 1370348857 | 886 | 2229607857 | 887 | 1314894769 | 888 | 2641504209 |
889 | 1094391521 | 890 | 3179094297 | 891 | 1421740409 | 892 | 2424853489 |
893 | 2003784313 | 894 | 2601422553 | 895 | 1324413313 | 896 | 3435753297 |
897 | 1457825417 | 898 | 3103542033 | 899 | 1602399217 | 900 | 4071247833 |
901 | 798447257 | 902 | 3530623273 | 903 | 1367841121 | 904 | 2494992417 |
905 | 1568185337 | 906 | 1507151409 | 907 | 1403055569 | 908 | 3180171369 |
909 | 720843289 | 910 | 3779810841 | 911 | 1697289089 | 912 | 2537937033 |
913 | 1444968089 | 914 | 3481002897 | 915 | 1492811897 | 916 | 3819843393 |
917 | 1201733153 | 918 | 2923319177 | 919 | 1520806369 | 920 | 3225574697 |
921 | 1219732793 | 922 | 3645265017 | 923 | 1480790041 | 924 | 2561150937 |
925 | 1303714201 | 926 | 2471502009 | 927 | 1672411249 | 928 | 3086409481 |
929 | 1134207353 | 930 | 3259442041 | 931 | 1342180313 | 932 | 3616278537 |
933 | 1823027209 | 934 | 3784692993 | 935 | 1818849121 | 936 | 3746518561 |
937 | 1919893561 | 938 | 4533867321 | 939 | 1944481873 | 940 | 2560471321 |
941 | 1902900761 | 942 | 2760382353 | 943 | 1998745937 | 944 | 4549819233 |
945 | 1727638009 | 946 | 2346199689 | 947 | 1791347633 | 948 | 3866224849 |
949 | 1776360337 | 950 | 2284988073 | 951 | 1859912113 | 952 | 3095542209 |
953 | 1968641401 | 954 | 4091400681 | 955 | 1242925193 | 956 | 4225706769 |
k | \(d_1\) | k | \(d_1\) | k | \(d_1\) | k | \(d_1\) |
---|---|---|---|---|---|---|---|
957 | 1892915281 | 958 | 4045337521 | 959 | 1343608489 | 960 | 3763601697 |
961 | 1814149921 | 962 | 4314724129 | 963 | 1530725993 | 964 | 3609639993 |
965 | 2095192361 | 966 | 2436844953 | 967 | 1351773289 | 968 | 3652908681 |
969 | 1829988737 | 970 | 3910470657 | 971 | 1267624417 | 972 | 3916603681 |
973 | 1608509857 | 974 | 3902482497 | 975 | 2267415617 | 976 | 4570911097 |
977 | 1857529337 | 978 | 4374788097 | 979 | 2397032977 | 980 | 5057369153 |
981 | 1670595169 | 982 | 4467133353 | 983 | 1846662313 | 984 | 4018295033 |
985 | 1762663633 | 986 | 3335288433 | 987 | 1284164017 | 988 | 4156050761 |
989 | 1285009081 | 990 | 4916862897 | 991 | 2476064449 | 992 | 4493035633 |
993 | 1271996969 | 994 | 4194691593 | 995 | 1940749649 | 996 | 4604658081 |
997 | 1711463473 | 998 | 4304953977 | 999 | 1545011873 | 1000 | 5834979417 |
1001 | 1847473897 | 1002 | 4000561161 | 1003 | 2587251313 | 1004 | 4444371537 |
1005 | 1970441177 | 1006 | 3405718369 | 1007 | 1382096873 | 1008 | 5178973193 |
1009 | 1676958233 | 1010 | 3411899313 | 1011 | 1764844217 | 1012 | 3933861297 |
1013 | 2665391921 | 1014 | 4085187033 | 1015 | 2155796977 | 1016 | 4976692977 |
1017 | 1727206529 | 1018 | 4202383529 | 1019 | 2238334649 | 1020 | 4709707473 |
1021 | 1906163689 | 1022 | 4994064329 | 1023 | 2099589449 | 1024 | 3672154313 |
1025 | 2398817921 | 1026 | 4792036593 | 1027 | 2531008121 | 1028 | 6935325609 |
1029 | 2054505073 | 1030 | 3141709761 | 1031 | 2776699313 | 1032 | 5715222753 |
1033 | 2134171513 | 1034 | 4244558761 | 1035 | 1927111289 | 1036 | 5127372177 |
1037 | 1838393969 | 1038 | 4227048849 | 1039 | 2140121833 | 1040 | 2998130601 |
1041 | 2288746241 | 1042 | 4985786569 | 1043 | 2278021241 | 1044 | 3622165017 |
1045 | 1723885337 | 1046 | 4033170177 | 1047 | 1948369009 | 1048 | 4866766521 |
1049 | 2181026153 | 1050 | 4926721809 | 1051 | 1899948697 | 1052 | 4513174257 |
1053 | 1780107257 | 1054 | 6008035881 | 1055 | 1981477217 | 1056 | 4505164217 |
1057 | 2128430041 | 1058 | 3772506593 | 1059 | 2822718281 | 1060 | 4274440017 |
1061 | 1704499681 | 1062 | 4783736721 | 1063 | 1541303833 | 1064 | 4850259177 |
1065 | 2098010809 | 1066 | 5239266033 | 1067 | 2637658841 | 1068 | 4392626921 |
1069 | 2149662329 | 1070 | 5038266297 | 1071 | 2158972121 | 1072 | 4546632633 |
1073 | 2086733353 | 1074 | 3829269273 | 1075 | 2998234721 | 1076 | 6655000377 |
1077 | 2648174233 | 1078 | 5672683977 | 1079 | 2252184889 | 1080 | 5032809849 |
1081 | 1771959913 | 1082 | 5685794529 | 1083 | 2053479553 | 1084 | 5816631921 |
1085 | 1511663233 | 1086 | 4288434441 | 1087 | 2787474721 | 1088 | 4952525601 |
1089 | 2244439553 | 1090 | 6708573953 | 1091 | 3044027353 | 1092 | 4837226241 |
1093 | 2746126073 | 1094 | 7094498513 | 1095 | 2877152249 | 1096 | 6873218697 |
1097 | 1772069249 | 1098 | 4869092089 | 1099 | 2817189961 | 1100 | 4743326977 |
1101 | 3022586257 | 1102 | 4954226601 | 1103 | 2498748409 | 1104 | 6842596993 |
1105 | 2487064193 | 1106 | 7970709993 | 1107 | 2353272241 | 1108 | 5216995617 |
1109 | 1926922913 | 1110 | 4832534281 | 1111 | 2792473897 | 1112 | 5454046977 |
1113 | 3116936497 | 1114 | 6769807377 | 1115 | 2517392177 | 1116 | 5703735801 |
1117 | 2671437473 | 1118 | 7489651409 | 1119 | 2477257033 | 1120 | 5843140473 |
1121 | 2207569633 | 1122 | 6967177233 | 1123 | 3071942201 | 1124 | 5210538337 |
1125 | 2860318481 | 1126 | 4725738361 | 1127 | 2158369337 | 1128 | 4380257841 |
1129 | 2955926249 | 1130 | 6026522217 | 1131 | 2854320649 | 1132 | 8687265337 |
1133 | 3132253633 | 1134 | 3941363721 | 1135 | 3007750849 | 1136 | 8538508113 |
1137 | 2660576873 | 1138 | 4614141057 | 1139 | 2714667961 | 1140 | 5295320529 |
1141 | 2520918889 | 1142 | 5026383321 | 1143 | 3100841593 | 1144 | 5145053313 |
1145 | 3892302041 | 1146 | 6648477441 | 1147 | 3307165321 | 1148 | 7713100497 |
k | \(d_1\) | k | \(d_1\) | k | \(d_1\) | k | \(d_1\) |
---|---|---|---|---|---|---|---|
1149 | 3459028681 | 1150 | 8423248593 | 1151 | 3855433433 | 1152 | 6303556081 |
1153 | 4081831337 | 1154 | 8181165073 | 1155 | 2370744913 | 1156 | 5505018033 |
1157 | 3265573513 | 1158 | 6115577073 | 1159 | 3420560993 | 1160 | 8620240729 |
1161 | 2074438369 | 1162 | 6685243161 | 1163 | 3091638289 | 1164 | 6461731329 |
1165 | 2557770689 | 1166 | 8386327569 | 1167 | 3909415849 | 1168 | 7717795649 |
1169 | 3346828609 | 1170 | 7157766009 | 1171 | 2546895473 | 1172 | 6856502937 |
1173 | 2793810361 | 1174 | 6047040057 | 1175 | 2157996097 | 1176 | 7838284001 |
1177 | 3455089697 | 1178 | 6647141153 | 1179 | 2842924577 | 1180 | 3590709753 |
1181 | 2476686001 | 1182 | 6121270401 | 1183 | 3243393473 | 1184 | 8151936081 |
1185 | 2750240993 | 1186 | 8253927273 | 1187 | 4210292057 | 1188 | 7049052497 |
1189 | 3727000777 | 1190 | 6917832177 | 1191 | 1880429633 | 1192 | 8525295417 |
1193 | 2945848097 | 1194 | 7813345929 | 1195 | 4223004449 | 1196 | 7625410321 |
1197 | 2935348321 | 1198 | 8133389889 | 1199 | 3453727913 | 1200 | 9033894193 |
1201 | 3900859073 | 1202 | 7929221217 | 1203 | 4421113777 | 1204 | 5713891737 |
1205 | 3245362609 | 1206 | 8728800369 | 1207 | 2985830417 | 1208 | 9863502889 |
1209 | 3775474961 | 1210 | 8046600153 | 1211 | 3447639473 | 1212 | 4400227329 |
1213 | 2407830617 | 1214 | 8811977777 | 1215 | 2882908073 | 1216 | 7791324753 |
1217 | 2283976033 | 1218 | 4647909473 | 1219 | 3339966881 | 1220 | 9564956097 |
1221 | 2923551601 | 1222 | 6811015929 | 1223 | 3830663849 | 1224 | 7028709697 |
1225 | 4063280401 | 1226 | 6696778449 | 1227 | 3844261441 | 1228 | 8576958993 |
1229 | 3535227961 | 1230 | 7620325329 | 1231 | 3092311897 | 1232 | 8908428153 |
1233 | 3776364689 | 1234 | 9188637753 | 1235 | 4696847321 | 1236 | 7724221737 |
1237 | 3856164217 | 1238 | 8480502209 | 1239 | 3861511657 | 1240 | 9144421801 |
1241 | 3395460017 | 1242 | 9294254817 | 1243 | 4180275881 | 1244 | 7201797177 |
1245 | 4078091713 | 1246 | 8112886233 | 1247 | 4499842297 | 1248 | 7881839529 |
1249 | 3775972633 | 1250 | 7019585673 | 1251 | 4105962433 | 1252 | 5848850001 |
1253 | 3991021849 | 1254 | 6065628801 | 1255 | 3564439913 | 1256 | 10740230169 |
1257 | 3463552753 | 1258 | 9561871929 | 1259 | 3185608033 | 1260 | 6169512273 |
1261 | 4293851609 | 1262 | 9078298377 | 1263 | 3156206177 | 1264 | 8582985849 |
1265 | 3897792881 | 1266 | 10075921769 | 1267 | 4139956313 | 1268 | 10380813601 |
1269 | 4570493801 | 1270 | 9356528801 | 1271 | 3890225513 | 1272 | 9442108113 |
1273 | 5438791777 | 1274 | 9367475217 | 1275 | 4870511089 | 1276 | 11355293009 |
1277 | 4570534433 | 1278 | 5457766953 | 1279 | 4582329937 | 1280 | 7966743801 |
1281 | 3303470393 | 1282 | 10603149393 | 1283 | 2858681489 | 1284 | 8329309041 |
1285 | 4838698793 | 1286 | 7925188569 | 1287 | 4214651369 | 1288 | 6365154713 |
1289 | 4383924977 | 1290 | 7700745369 | 1291 | 4783441193 | 1292 | 10899763737 |
1293 | 3183701393 | 1294 | 7947641353 | 1295 | 4216864601 | 1296 | 10375292481 |
1297 | 4884197377 | 1298 | 8881826633 | 1299 | 3896076089 | 1300 | 12504176953 |
1301 | 4842107689 | 1302 | 5756693721 | 1303 | 2723480233 | 1304 | 10029103001 |
1305 | 5075648857 | 1306 | 10231827369 | 1307 | 4881446321 | 1308 | 10362529857 |
1309 | 4663594841 | 1310 | 8630540241 | 1311 | 4361721577 | 1312 | 6785253561 |
1313 | 4473287153 | 1314 | 11339103273 | 1315 | 5213216977 | 1316 | 9030929009 |
1317 | 3765257761 | 1318 | 8397845481 | 1319 | 5921712193 | 1320 | 10518869337 |
1321 | 3553813609 | 1322 | 10411769897 | 1323 | 4880881361 | 1324 | 6586436201 |
1325 | 4041084001 | 1326 | 9637389353 | 1327 | 4263481897 | 1328 | 10794914769 |
1329 | 4020541969 | 1330 | 8861462817 | 1331 | 4655094673 | 1332 | 10320158377 |
1333 | 4225091729 | 1334 | 12687213841 | 1335 | 4849883897 | 1336 | 8635792713 |
1337 | 5056156097 | 1338 | 11247364209 | 1339 | 3533357441 | 1340 | 12338265921 |
k | \(d_1\) | k | \(d_1\) | k | \(d_1\) | k | \(d_1\) |
---|---|---|---|---|---|---|---|
1341 | 6188174921 | 1342 | 10179842689 | 1343 | 4931635369 | 1344 | 10717061457 |
1345 | 4992350177 | 1346 | 9914338641 | 1347 | 3640667057 | 1348 | 10149929057 |
1349 | 5178899449 | 1350 | 11908911953 | 1351 | 3790717769 | 1352 | 9688139169 |
1353 | 2687142617 | 1354 | 11319565489 | 1355 | 5446670857 | 1356 | 10184514657 |
1357 | 4413137921 | 1358 | 11811948801 | 1359 | 5491510897 | 1360 | 14981176697 |
1361 | 6845363297 | 1362 | 11520637489 | 1363 | 5726707801 | 1364 | 9844492793 |
1365 | 4074339769 | 1366 | 13189267281 | 1367 | 4590877273 | 1368 | 10263989121 |
1369 | 6090656801 | 1370 | 11577597441 | 1371 | 5360341369 | 1372 | 8969163897 |
1373 | 3729738193 | 1374 | 10030041849 | 1375 | 5790874417 | 1376 | 13672932177 |
1377 | 3354967201 | 1378 | 9721180921 | 1379 | 3593612993 | 1380 | 11572057281 |
1381 | 2975326049 | 1382 | 11990306353 | 1383 | 5409192193 | 1384 | 12071006817 |
1385 | 6181709449 | 1386 | 14069494849 | 1387 | 6109066921 | 1388 | 11024766993 |
1389 | 5274319777 | 1390 | 9642295441 | 1391 | 6249679649 | 1392 | 11173712089 |
1393 | 3920782057 | 1394 | 14970384881 | 1395 | 4291103633 | 1396 | 14873830449 |
1397 | 4970225489 | 1398 | 14282140161 | 1399 | 3113018473 | 1400 | 9907899873 |
1401 | 5602055801 | 1402 | 13224194153 | 1403 | 4388529953 | 1404 | 9080218569 |
1405 | 5874339553 | 1406 | 14392574233 | 1407 | 5313687361 | 1408 | 8070174249 |
1409 | 5340878497 | 1410 | 8445957249 | 1411 | 5799147161 | 1412 | 11307387921 |
1413 | 6256545641 | 1414 | 14603912313 | 1415 | 5182399633 | 1416 | 11291079633 |
1417 | 5761864793 | 1418 | 8611063521 | 1419 | 4283780297 | 1420 | 11965703817 |
1421 | 4919896369 | 1422 | 13148498001 | 1423 | 5418737449 | 1424 | 11819709609 |
1425 | 6452481929 | 1426 | 8849047089 | 1427 | 6464797601 | 1428 | 10724557281 |
1429 | 5850358097 | 1430 | 11668121193 | 1431 | 4450743881 | 1432 | 14117182377 |
1433 | 7530295477 | 1434 | 11005822041 | 1435 | 4781099249 | 1436 | 16464319257 |
1437 | 5625246553 | 1438 | 15200800833 | 1439 | 6232068121 | 1440 | 11543895273 |
1441 | 5582017241 | 1442 | 11467212681 | 1443 | 6162816449 | 1444 | 12587849569 |
1445 | 4505553209 | 1446 | 9179582801 | 1447 | 4759466801 | 1448 | 12803192537 |
1449 | 5190934153 | 1450 | 13376175873 | 1451 | 5026362641 | 1452 | 17479263937 |
1453 | 6196053953 | 1454 | 12803941329 | 1455 | 6038713553 | 1456 | 14943995697 |
1457 | 5826184769 | 1458 | 12952966161 | 1459 | 8336832073 | 1460 | 12575735841 |
1461 | 6697005209 | 1462 | 12862360281 | 1463 | 6953992769 | 1464 | 14337419209 |
1465 | 5551779049 | 1466 | 13519359177 | 1467 | 5740147537 | 1468 | 11113911169 |
1469 | 6633294233 | 1470 | 10581863009 | 1471 | 6089924993 | 1472 | 14453700969 |
1473 | 6641047097 | 1474 | 13050281721 | 1475 | 6676376873 | 1476 | 13635058089 |
1477 | 4879320689 | 1478 | 11540902641 | 1479 | 5218748209 | 1480 | 15361738681 |
1481 | 5142288889 | 1482 | 8955885801 | 1483 | 8016755057 | 1484 | 12068317977 |
1485 | 5782637041 | 1486 | 14976660297 | 1487 | 7880353297 | 1488 | 16452839881 |
1489 | 6484799473 | 1490 | 9905393521 | 1491 | 5707371641 | 1492 | 17674712841 |
1493 | 6689379689 | 1494 | 12127943433 | 1495 | 6879464849 | 1496 | 16182669297 |
1497 | 4306434697 | 1498 | 14691262881 | 1499 | 8152320377 | 1500 | 15122451873 |
1501 | 6323595697 | 1502 | 14457595441 | 1503 | 6009027289 | 1504 | 13185180849 |
1505 | 5685159793 | 1506 | 14022943169 | 1507 | 8097404209 | 1508 | 16948680681 |
1509 | 6547557281 | 1510 | 14908391969 | 1511 | 7678220273 | 1512 | 19449660561 |
1513 | 6075080353 | 1514 | 17372940729 | 1515 | 6823802473 | 1516 | 10667702049 |
1517 | 7286535961 | 1518 | 16155325929 | 1519 | 6453592841 | 1520 | 14756469729 |
1521 | 7044532081 | 1522 | 11995984617 | 1523 | 6981504113 | 1524 | 17180091273 |
1525 | 7773449641 | 1526 | 19556487329 | 1527 | 6239985569 | 1528 | 18678344721 |
1529 | 7319054153 | 1530 | 9633634761 | 1531 | 8359447409 | 1532 | 19243693401 |
k | \(d_1\) | k | \(d_1\) | k | \(d_1\) | k | \(d_1\) |
---|---|---|---|---|---|---|---|
1533 | 5383103641 | 1534 | 15336632257 | 1535 | 5435424889 | 1536 | 16518110361 |
1537 | 6493576057 | 1538 | 11799669513 | 1539 | 6261258617 | 1540 | 11453876897 |
1541 | 7007402449 | 1542 | 16811621457 | 1543 | 7119113321 | 1544 | 15084189201 |
1545 | 5032367849 | 1546 | 19853096433 | 1547 | 6206696401 | 1548 | 18095827017 |
1549 | 5585570233 | 1550 | 12228799353 | 1551 | 8651999977 | 1552 | 13545865481 |
1553 | 7061734417 | 1554 | 15351766761 | 1555 | 7881434881 | 1556 | 19088671361 |
1557 | 6358784129 | 1558 | 17521254633 | 1559 | 8866491401 | 1560 | 14777854617 |
1561 | 5406250657 | 1562 | 16055379201 | 1563 | 8111629553 | 1564 | 18455413729 |
1565 | 7557590969 | 1566 | 19354854017 | 1567 | 6376897561 | 1568 | 11162316201 |
1569 | 10325243801 | 1570 | 14174249937 | 1571 | 8346350809 | 1572 | 14805992793 |
1573 | 8639085097 | 1574 | 19876647729 | 1575 | 7310427353 | 1576 | 13383109041 |
1577 | 7692909481 | 1578 | 17923856217 | 1579 | 7980327521 | 1580 | 13791208929 |
1581 | 4445597953 | 1582 | 14074036089 | 1583 | 5656117361 | 1584 | 17916753801 |
1585 | 9527118401 | 1586 | 14217349353 | 1587 | 7082431481 | 1588 | 16185164297 |
1589 | 6205767769 | 1590 | 16566006801 | 1591 | 9414671033 | 1592 | 19394644593 |
1593 | 9484650257 | 1594 | 10596671121 | 1595 | 8488474417 | 1596 | 13021079817 |
1597 | 7823658209 | 1598 | 24921801969 | 1599 | 9490053809 | 1600 | 19530979913 |
1601 | 9941854033 | 1602 | 11917095513 | 1603 | 8980051961 | 1604 | 17863582801 |
1605 | 8408412017 | 1606 | 19317873369 | 1607 | 9030034649 | 1608 | 21150606513 |
1609 | 10040135537 | 1610 | 22641182337 | 1611 | 7404361369 | 1612 | 18619595009 |
1613 | 11724971569 | 1614 | 14844290169 | 1615 | 8054991409 | 1616 | 18427052337 |
1617 | 7654546177 | 1618 | 20158634841 | 1619 | 8074096649 | 1620 | 10101987681 |
1621 | 6833900393 | 1622 | 21247408473 | 1623 | 9205065929 | 1624 | 19442890041 |
1625 | 9320044529 | 1626 | 20777329497 | 1627 | 7474670633 | 1628 | 16407640929 |
1629 | 8062549801 | 1630 | 19745737113 | 1631 | 5708233177 | 1632 | 19947287721 |
1633 | 7857319273 | 1634 | 24913676649 | 1635 | 10897495537 | 1636 | 18635575281 |
1637 | 9765563569 | 1638 | 22715248593 | 1639 | 6518086921 | 1640 | 11710935681 |
1641 | 10877325889 | 1642 | 24782078073 | 1643 | 7997553217 | 1644 | 17307309497 |
1645 | 9688815713 | 1646 | 20375827873 | 1647 | 8266590337 | 1648 | 21286015017 |
1649 | 6771151313 | 1650 | 21418425473 | 1651 | 7759261313 | 1652 | 17058280017 |
1653 | 7055863681 | 1654 | 23824112489 | 1655 | 7797935281 | 1656 | 22941611697 |
1657 | 6898160657 | 1658 | 19080932649 | 1659 | 8573955281 | 1660 | 21767826777 |
1661 | 11803489417 | 1662 | 20306822649 | 1663 | 10557367441 | 1664 | 19351782129 |
1665 | 9643518041 | 1666 | 23188938609 | 1667 | 8053982201 | 1668 | 17951834217 |
1669 | 9542343233 | 1670 | 26324496353 | 1671 | 8320139033 | 1672 | 14965707817 |
1673 | 9223372409 | 1674 | 16030498793 | 1675 | 12397060721 | 1676 | 21302932753 |
1677 | 7388864993 | 1678 | 20675197713 | 1679 | 8424181121 | 1680 | 15850391313 |
1681 | 9979688393 | 1682 | 23169114809 | 1683 | 9821011049 | 1684 | 21532687521 |
1685 | 9032961017 | 1686 | 17768279433 | 1687 | 8223478961 | 1688 | 19766229081 |
1689 | 12157301161 | 1690 | 18187374489 | 1691 | 10753068737 | 1692 | 18319821657 |
1693 | 9909960329 | 1694 | 26639467017 | 1695 | 10258117313 | 1696 | 21471116241 |
1697 | 13692281329 | 1698 | 15424707057 | 1699 | 6648674609 | 1700 | 22105438041 |
1701 | 10328316337 | 1702 | 23340311481 | 1703 | 7672509353 | 1704 | 19865714313 |
1705 | 7345957961 | 1706 | 20083328049 | 1707 | 12110334193 | 1708 | 12603492609 |
1709 | 14665471217 | 1710 | 21153612153 | 1711 | 12214804297 | 1712 | 16901015217 |
1713 | 7883012129 | 1714 | 20265748617 | 1715 | 11526296753 | 1716 | 17900661153 |
1717 | 9022814057 | 1718 | 23173716129 | 1719 | 8558039537 | 1720 | 18317261649 |
1721 | 9755539897 | 1722 | 26642566113 | 1723 | 11295454553 | 1724 | 21852465793 |
k | \(d_1\) | k | \(d_1\) | k | \(d_1\) | k | \(d_1\) |
---|---|---|---|---|---|---|---|
1725 | 9820146593 | 1726 | 25820071817 | 1727 | 13386871657 | 1728 | 19234668121 |
1729 | 10200354113 | 1730 | 22356100321 | 1731 | 10496099857 | 1732 | 15563628777 |
1733 | 10280518337 | 1734 | 19458436089 | 1735 | 10344401849 | 1736 | 15334875609 |
1737 | 12478800689 | 1738 | 22729799577 | 1739 | 13185137881 | 1740 | 31556733609 |
1741 | 11977301593 | 1742 | 13991205561 | 1743 | 9726851417 | 1744 | 26735056217 |
1745 | 7959615929 | 1746 | 20732047473 | 1747 | 13544803529 | 1748 | 23934761337 |
1749 | 10109738497 | 1750 | 15138620841 | 1751 | 10531936889 | 1752 | 26307135033 |
1753 | 11742293489 | 1754 | 15293415993 | 1755 | 11740768681 | 1756 | 21444330777 |
1757 | 11408438473 | 1758 | 26855399481 | 1759 | 9561059017 | 1760 | 19279833721 |
1761 | 9953890057 | 1762 | 25903870689 | 1763 | 13747239089 | 1764 | 23244127729 |
1765 | 10512541913 | 1766 | 29607980417 | 1767 | 14566400561 | 1768 | 31222130153 |
1769 | 13852180417 | 1770 | 29675608953 | 1771 | 9052924193 | 1772 | 21842571921 |
1773 | 12653890049 | 1774 | 21060861537 | 1775 | 10345905937 | 1776 | 22913920929 |
1777 | 11299193849 | 1778 | 23773736001 | 1779 | 12959578369 | 1780 | 27545395593 |
1781 | 11502866521 | 1782 | 27768982497 | 1783 | 12903652081 | 1784 | 28861680633 |
1785 | 12271976993 | 1786 | 24459731697 | 1787 | 10862200793 | 1788 | 15961497577 |
1789 | 10612904713 | 1790 | 24324206537 | 1791 | 10638082801 | 1792 | 31848889713 |
1793 | 10500588257 |
Selected values \(1795 \leqslant k \leqslant 2941\) are also considered.
k | \(d_1\) | k | \(d_1\) | k | \(d_1\) | k | \(d_1\) |
---|---|---|---|---|---|---|---|
1795 | 14302640609 | 1796 | 27358359081 | 1797 | 11675814881 | 1798 | 27273950169 |
1799 | 12288253057 | 1800 | 29409340097 | 1801 | 10747223609 | 1802 | 17116275873 |
1803 | 10761662033 | 1805 | 11795108209 | 1806 | 24360921033 | 1807 | 14006501017 |
1809 | 11315214497 | 1810 | 30452439257 | 1811 | 10525057129 | 1812 | 28348764753 |
1813 | 8922565193 | 1814 | 31795855249 | 1815 | 10367096249 | 1816 | 28742795673 |
1817 | 8638659049 | 1818 | 24484322193 | 1819 | 13819473449 | 1820 | 15894922737 |
1821 | 11682471617 | 1822 | 28719137049 | 1823 | 10534815857 | 1824 | 17666542041 |
1825 | 10081327513 | 1827 | 12077934553 | 1828 | 22366484121 | 1829 | 14616252913 |
1830 | 20798720121 | 1831 | 11063929649 | 1832 | 28289473809 | 1833 | 8002632881 |
1834 | 25611350433 | 1835 | 9212193857 | 1836 | 27665418993 | 1837 | 16898429161 |
1838 | 26484256857 | 1839 | 9016016953 | 1840 | 24896737497 | 1841 | 15427422041 |
1842 | 26136546297 | 1843 | 10786881689 | 1844 | 28223186289 | 1845 | 16636782473 |
1847 | 12676735273 | 1849 | 7747069097 | 1850 | 28252003081 | 1851 | 10879109849 |
1852 | 26361529041 | 1853 | 13897278121 | 1854 | 21989559201 | 1855 | 10695553697 |
1856 | 27284626329 | 1857 | 14003640817 | 1858 | 27657269313 | 1859 | 12177360529 |
1860 | 21356618529 | 1861 | 11888816113 | 1862 | 30328356601 | 1863 | 9906836593 |
1864 | 21976435497 | 1865 | 14472946801 | 1866 | 20748624513 | 1867 | 11552645537 |
1868 | 27402172977 | 1869 | 11303255617 | 1870 | 24913897737 | 1871 | 8443601753 |
1872 | 31756304409 | 1873 | 16026449393 | 1874 | 29842152657 | 1875 | 12307826081 |
1876 | 27713889569 | 1877 | 9654472721 | 1879 | 15210808849 | 1880 | 28870006929 |
1881 | 15669411673 | 1882 | 28934387553 | 1883 | 16555237537 | 1885 | 11660985689 |
1886 | 27292507377 | 1887 | 14953069561 | 1889 | 10128518657 | 1891 | 17239580153 |
1892 | 30427087497 | 1893 | 11212691801 | 1894 | 30501258393 | 1895 | 15299539457 |
1896 | 27590751057 | 1897 | 13077468809 | 1898 | 31827861641 | 1899 | 11483109641 |
1901 | 12606717017 | 1903 | 15584020681 | 1905 | 11813267161 | 1906 | 16367835009 |
1907 | 18412667969 | 1909 | 11836986241 | 1910 | 21635667673 | 1911 | 13272833377 |
1912 | 30472364553 | 1913 | 12531171217 | 1914 | 23353861521 | 1915 | 8202143393 |
k | \(d_1\) | k | \(d_1\) | k | \(d_1\) | k | \(d_1\) |
---|---|---|---|---|---|---|---|
1917 | 14755875137 | 1918 | 25708307601 | 1919 | 15066358481 | 1920 | 22043395081 |
1921 | 9023034793 | 1923 | 11690362561 | 1925 | 14786098913 | 1927 | 13989520417 |
1929 | 10757106673 | 1930 | 26996413497 | 1931 | 11823613913 | 1933 | 16867973321 |
1935 | 15576283369 | 1937 | 16795975817 | 1939 | 17334712729 | 1941 | 15871885753 |
1942 | 31072472337 | 1943 | 12023446697 | 1945 | 10890996689 | 1946 | 22747002657 |
1947 | 16774867913 | 1948 | 27535083801 | 1949 | 14984974273 | 1950 | 23125134057 |
1951 | 15365245153 | 1952 | 28463088977 | 1953 | 14557374529 | 1954 | 29968558449 |
1955 | 16991135209 | 1956 | 29772869289 | 1957 | 18647799593 | 1958 | 30016450209 |
1959 | 17536828609 | 1961 | 16387251809 | 1963 | 16939173761 | 1964 | 24017100321 |
1965 | 17519090473 | 1966 | 25475162953 | 1967 | 18063913321 | 1969 | 19189797049 |
1971 | 16094600017 | 1973 | 19185850561 | 1975 | 22692362201 | 1977 | 13792212017 |
1979 | 16926239609 | 1981 | 18654590113 | 1982 | 31863559753 | 1983 | 11258242889 |
1984 | 26625714369 | 1985 | 17771644441 | 1987 | 19770734129 | 1989 | 16941181249 |
1991 | 16964769281 | 1993 | 18166946593 | 1995 | 19893527657 | 1997 | 10175019889 |
1999 | 18426841921 | 2001 | 16120899073 | 2003 | 10683271289 | 2005 | 19279072217 |
2007 | 20008876177 | 2009 | 15135767321 | 2011 | 12772102033 | 2013 | 15387373321 |
2015 | 17258610449 | 2016 | 26923237881 | 2017 | 15488878849 | 2019 | 20066054209 |
2021 | 14898157433 | 2022 | 25753502769 | 2023 | 19001943929 | 2025 | 17314824481 |
2027 | 11829608209 | 2029 | 16940925433 | 2031 | 19334270129 | 2033 | 23685116761 |
2035 | 18311081017 | 2037 | 14836645081 | 2039 | 16373229137 | 2041 | 17298071521 |
2043 | 14748148673 | 2045 | 10982533169 | 2047 | 21082442393 | 2049 | 23866962377 |
2051 | 18769206193 | 2053 | 19125553481 | 2054 | 29534671497 | 2055 | 14107004761 |
2057 | 21482977393 | 2059 | 14756819873 | 2061 | 12912128689 | 2063 | 10837182401 |
2065 | 18331106561 | 2067 | 10481469337 | 2069 | 17415046001 | 2071 | 20282274193 |
2073 | 18819832793 | 2075 | 17545036193 | 2077 | 15578650289 | 2078 | 27729367257 |
2079 | 24057117089 | 2081 | 16148176609 | 2082 | 25882365777 | 2083 | 12490953649 |
2085 | 10832312753 | 2087 | 19455168137 | 2089 | 18270610081 | 2091 | 21856626049 |
2093 | 23238238169 | 2095 | 24068136049 | 2097 | 16522756057 | 2099 | 15962858257 |
2101 | 25476168961 | 2103 | 24847607321 | 2105 | 18015016057 | 2106 | 31923588929 |
2107 | 20803781729 | 2109 | 19673142289 | 2111 | 18392434289 | 2113 | 23220719273 |
2115 | 12636915289 | 2117 | 19315124009 | 2119 | 28994112073 | 2121 | 20050974761 |
2123 | 29088481481 | 2125 | 18655061473 | 2127 | 19574918641 | 2129 | 18895809337 |
2131 | 23112080113 | 2133 | 14523603209 | 2135 | 22249389529 | 2137 | 21773555593 |
2139 | 18025003393 | 2141 | 22630967561 | 2142 | 29046167529 | 2143 | 18693988129 |
2145 | 18877428953 | 2147 | 18136409801 | 2149 | 19695859753 | 2151 | 19861422377 |
2153 | 24196460777 | 2155 | 23555454961 | 2156 | 29713956801 | 2157 | 20859254233 |
2159 | 20632640297 | 2161 | 27332078161 | 2162 | 28975285761 | 2163 | 18060492217 |
2165 | 21744616193 | 2167 | 20857806913 | 2169 | 14622455329 | 2171 | 20886550177 |
2173 | 14434855249 | 2175 | 16846410769 | 2177 | 15977018641 | 2179 | 23282095961 |
2181 | 25607323801 | 2183 | 26851308977 | 2185 | 19995487417 | 2187 | 26886280081 |
2189 | 27014177657 | 2191 | 14464864153 | 2193 | 20535271249 | 2195 | 26841314281 |
2197 | 22297034849 | 2199 | 23926888417 | 2201 | 23201739361 | 2203 | 16987173097 |
2205 | 18142454081 | 2207 | 16398494993 | 2209 | 27148857433 | 2211 | 22374953593 |
2213 | 23897368169 | 2215 | 20780753537 | 2217 | 24137876041 | 2219 | 21973542001 |
2221 | 21305351633 | 2222 | 31449231969 | 2223 | 25203670673 | 2224 | 31563684681 |
2225 | 19548976033 | 2227 | 19151122577 | 2229 | 17388825793 | 2231 | 20015710609 |
2233 | 23774802841 | 2235 | 17229928721 | 2237 | 25054037273 | 2239 | 28215927697 |
k | \(d_1\) | k | \(d_1\) | k | \(d_1\) | k | \(d_1\) |
---|---|---|---|---|---|---|---|
2241 | 25364314097 | 2243 | 24693406769 | 2245 | 19663680161 | 2247 | 23166334049 |
2249 | 22198449089 | 2251 | 21782856049 | 2253 | 23058697577 | 2255 | 23490722401 |
2257 | 23182069777 | 2259 | 25375332953 | 2261 | 23321397889 | 2263 | 23134466353 |
2265 | 21099281657 | 2267 | 20500043689 | 2269 | 24136990729 | 2271 | 26367890993 |
2273 | 23874674297 | 2275 | 18425670833 | 2277 | 26640229873 | 2279 | 23580652217 |
2281 | 18777503057 | 2282 | 28804400601 | 2283 | 25006505753 | 2285 | 28135395473 |
2287 | 25885481009 | 2289 | 24003072217 | 2291 | 25217386561 | 2293 | 22590552449 |
2295 | 24478979353 | 2297 | 29279333513 | 2299 | 21622666817 | 2301 | 28206525689 |
2303 | 29479981529 | 2305 | 30915727681 | 2307 | 23347446401 | 2309 | 30173417497 |
2311 | 17259667001 | 2313 | 20830862281 | 2315 | 28875641633 | 2317 | 22175499113 |
2319 | 20768662297 | 2321 | 22111529257 | 2323 | 22985300057 | 2325 | 21970530497 |
2327 | 21542492929 | 2329 | 25253712697 | 2331 | 24055982809 | 2333 | 27344154281 |
2335 | 31659517921 | 2337 | 26380992137 | 2339 | 24173328793 | 2341 | 23702333329 |
2343 | 22874829473 | 2345 | 27503106937 | 2347 | 23997306689 | 2349 | 25080505033 |
2351 | 27005350529 | 2353 | 19594199089 | 2355 | 26290364593 | 2357 | 31187585617 |
2361 | 21535123417 | 2363 | 30890690297 | 2365 | 17997494969 | 2367 | 25941664313 |
2368 | 30521729001 | 2369 | 21152347649 | 2373 | 28640771921 | 2375 | 25445748593 |
2377 | 24642382537 | 2379 | 27949170833 | 2383 | 29346656233 | 2385 | 27353459369 |
2387 | 22092862993 | 2389 | 26541069889 | 2391 | 27922751849 | 2395 | 25679645297 |
2397 | 25567788161 | 2401 | 26806417097 | 2403 | 25471907233 | 2411 | 23201469721 |
2413 | 27471244057 | 2415 | 28729140457 | 2417 | 31918397593 | 2423 | 19416040537 |
2427 | 27250644433 | 2429 | 28702862873 | 2431 | 18452796697 | 2433 | 31076018153 |
2437 | 19044233393 | 2443 | 28724687897 | 2447 | 24087157561 | 2449 | 24340659377 |
2451 | 24310203641 | 2453 | 29613412849 | 2455 | 25877124769 | 2457 | 20105114921 |
2459 | 31572674153 | 2465 | 31474193953 | 2469 | 31751925329 | 2473 | 25877913169 |
2479 | 25781498417 | 2483 | 30687948241 | 2485 | 29489657473 | 2487 | 23214266969 |
2489 | 24403608241 | 2493 | 28496723993 | 2495 | 26242884937 | 2499 | 29141913769 |
2511 | 27983986649 | 2523 | 31630888169 | 2531 | 30568914073 | 2533 | 29836994353 |
2545 | 30815861849 | 2551 | 29163166121 | 2553 | 18839920273 | 2555 | 25973619241 |
2561 | 27573697457 | 2565 | 20167085041 | 2567 | 30338840489 | 2575 | 24854975473 |
2579 | 29969335969 | 2623 | 29674805977 | 2627 | 26057264561 | 2645 | 22700098081 |
2667 | 26463497129 | 2683 | 28569879721 | 2705 | 30513902753 | 2713 | 29668713889 |
2735 | 28004847841 | 2757 | 20013907409 | 2783 | 31014739937 | 2801 | 31532536313 |
2851 | 25306669001 | 2869 | 30730146737 | 2941 | 28715939033 |
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Dąbrowski, A., Jędrzejak, T., Szymaszkiewicz, L. (2016). Behaviour of the Order of Tate–Shafarevich Groups for the Quadratic Twists of \(X_0(49)\) . In: Loeffler, D., Zerbes, S. (eds) Elliptic Curves, Modular Forms and Iwasawa Theory. JHC70 2015. Springer Proceedings in Mathematics & Statistics, vol 188. Springer, Cham. https://doi.org/10.1007/978-3-319-45032-2_4
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