jsMath
Worksheet: kolykato

# The Explicit Upper Bounds of Kolyvagin and Kato

 `Elliptic Curve defined by y^2 + y = x^3 + x^2 - 12*x + 2 over Rational Field` `Elliptic Curve defined by y^2 + y = x^3 + x^2 - 12*x + 2 over Rational Field`

## Kato's Explicit Bound

Refining work of Kato on Euler systems involving K -groups of elliptic curves, we obtain the following theorem:

 Refined Kato Bound Let E  be an elliptic curve with L(E;1)==0 . Then #Sha(E)ÔCÁÊEL(E;1)ÁQcv(#E(Q)tors)2; where C  is divisible by 2  and primes of additive reduction and primes for which ÚE;p  is neither surjective nor reducible. (Wuthrich proved last year that if ÚE;p  is reducible then the BSD upper bound holds at p !)

For our curve 141a, the above theorem yields no information since the analytic rank of our curve is 1.
 `0.000000000000000` `0.000000000000000`
 `1` `1`
 `False` `False`

Incidentally, SAGE has a non-surjective command which gives all primes for which ÚE;p  is not surjective, and why. It uses results of Cojocaru, Kani, and Serre. This surjectivity hypothesis is important in many of theorems we will discuss.
 `[]` `[]`
 `[(2, '2-torsion'), (3, '3-torsion')]` `[(2, '2-torsion'), (3, '3-torsion')]`

## Kolyvagin's Explicit Bound

 Refined Version of Kolyvagin's Theorem Suppose E  is a non-CM elliptic curve over Q . Suppose K  is a quadratic imaginary field that satisfies the Heegner hypothesis and p  is an odd prime such that p=j#E0(K)tor  for any curve E0  that is Q -isogenous to E . Then ordp(#Sha(E))Ô2ordp([E(K):ZyK]);  unless disc(K)  is divisible by exactly one prime ` , in which case the conclusion is only valid if p==` .

Kolyvagin's theorem tells us that #Sha(E)  divides 49 . This is a huge statement: it tells us that either #Sha(E)=1  or #Sha(E)=49 .
 `([2, 7], 49)` `([2, 7], 49)`
The output of the shabound_kolyvagin command is a list of primes that could possibly divide #Sha(E) , followed by the square of odd part of the index [E(K):ZyK] .

Conclusion: Either #Sha(E)=1  or #Sha(E)=49 .

Even using more Heegner discriminants doesn't help. In fact the BSD conjecture implies that they are all divisible by 7 :
 `[-11, -20, -23, -35]` `[-11, -20, -23, -35]`
 ```[48.999572 ... 49.000367] [48.999633 ... 49.000306] [48.999572 ... 49.000306] [48.999633 ... 49.000367]``` ```[48.999572 ... 49.000367] [48.999633 ... 49.000306] [48.999572 ... 49.000306] [48.999633 ... 49.000367]```

## Kolyvagin's Method does get us pretty far though...

 ```11a1 ([2, 5], 1) [5] 14a1 ([2, 3], 1) [2, 3] 15a1 ([2], 1) [2, 4] 17a1 ([2], 1) [4] 19a1 ([2, 3], 1) [3] 20a1 ([2, 3], 1) [3, 2] 21a1 ([2], 1) [4, 2] 24a1 ([2], 1) [4, 2] 26a1 ([2, 3], 1) [1, 3] 26b1 ([2, 7], 1) [7, 1] 27a1 (0, 0) [3] 30a1 ([2, 3], 1) [2, 3, 1] 32a1 (0, 0) [4] 33a1 ([2], 1) [2, 2] 34a1 ([2, 3], 1) [6, 1] 35a1 ([2, 3], 1) [1, 3] 36a1 (0, 0) [3, 2] 37a1 ([2], 1) [1] 37b1 ([2, 3], 1) [3] 38a1 ([2, 3], 1) [1, 3] 38b1 ([2, 5], 1) [5, 1] 39a1 ([2], 1) [2, 2] 40a1 ([2], 1) [2, 2] 42a1 ([2], 1) [8, 2, 1] 43a1 ([2], 1) [1] 44a1 ([2, 3], 1) [3, 1] 45a1 ([2], 1) [2, 1] 46a1 ([2], 1) [2, 1] 48a1 ([2], 1) [2, 2] 49a1 (0, 0) [2] 50a1 ([2, 3, 5], 1) [1, 3] 50b1 ([2, 3, 5], 1) [5, 1] 51a1 ([2, 3], 1) [3, 1] 52a1 ([2], 1) [1, 2] 53a1 ([2], 1) [1] 54a1 ([2, 3], 1) [1, 3] 54b1 ([2, 3], 1) [3, 1] 55a1 ([2], 1) [2, 2] 56a1 ([2], 1) [4, 1] 56b1 ([2], 1) [2, 1] 57a1 ([2], 1) [2, 1] 57c1 ([2, 5], 1) [10, 1] 57b1 ([2], 1) [2, 2] 58a1 ([2], 1) [2, 1] 58b1 ([2, 5], 1) [10, 1] 61a1 ([2], 1) [1] 62a1 ([2], 1) [4, 1] 63a1 ([2], 1) [2, 1] 64a1 (0, 0) [4] 65a1 ([2], 1) [1, 1] 66a1 ([2, 3], 1) [2, 3, 1] 66c1 ([2, 5], 25) [10, 5, 1] 66b1 ([2], 1) [4, 1, 1] 67a1 ([2], 1) [1] 69a1 ([2], 1) [2, 1] 70a1 ([2], 1) [4, 2, 1] 72a1 ([2], 1) [2, 4] 73a1 ([2], 1) [2] 75a1 ([2, 5], 1) [1, 1] 75c1 ([2, 5], 1) [5, 1] 75b1 ([2], 1) [1, 2] 76a1 ([2], 1) [1, 1] 77a1 ([2], 1) [2, 1] 77c1 ([2], 1) [1, 2] 77b1 ([2, 3], 1) [6, 1] 78a1 ([2], 1) [2, 1, 1] 79a1 ([2], 1) [1] 80a1 ([2], 1) [2, 2] 80b1 ([2, 3], 1) [1, 2] 82a1 ([2], 1) [2, 1] 83a1 ([2], 1) [1] 84a1 ([2, 3], 9) [3, 3, 2] 84b1 ([2], 1) [1, 1, 2] 85a1 ([2], 1) [2, 1] 88a1 ([2], 1) [4, 1] 89a1 ([2], 1) [1] 89b1 ([2], 1) [2] 90a1 ([2, 3], 1) [2, 2, 3] 90c1 ([2, 3], 1) [4, 4, 1] 90b1 ([2, 3], 1) [6, 2, 1] 91a1 ([2], 1) [1, 1] 91b1 ([2, 3], 1) [1, 1] 92a1 ([2, 3], 1) [3, 1] 92b1 ([2, 3], 9) [3, 1] 94a1 ([2], 1) [2, 1] 96a1 ([2], 1) [2, 2] 96b1 ([2], 1) [2, 2] 98a1 ([2, 3], 1) [2, 2] 99a1 ([2], 1) [2, 1] 99c1 ([2], 1) [2, 1] 99b1 ([2], 1) [4, 1] 99d1 ([2, 5], 1) [1, 1]``` ```11a1 ([2, 5], 1) [5] 14a1 ([2, 3], 1) [2, 3] 15a1 ([2], 1) [2, 4] 17a1 ([2], 1) [4] 19a1 ([2, 3], 1) [3] 20a1 ([2, 3], 1) [3, 2] 21a1 ([2], 1) [4, 2] 24a1 ([2], 1) [4, 2] 26a1 ([2, 3], 1) [1, 3] 26b1 ([2, 7], 1) [7, 1] 27a1 (0, 0) [3] 30a1 ([2, 3], 1) [2, 3, 1] 32a1 (0, 0) [4] 33a1 ([2], 1) [2, 2] 34a1 ([2, 3], 1) [6, 1] 35a1 ([2, 3], 1) [1, 3] 36a1 (0, 0) [3, 2] 37a1 ([2], 1) [1] 37b1 ([2, 3], 1) [3] 38a1 ([2, 3], 1) [1, 3] 38b1 ([2, 5], 1) [5, 1] 39a1 ([2], 1) [2, 2] 40a1 ([2], 1) [2, 2] 42a1 ([2], 1) [8, 2, 1] 43a1 ([2], 1) [1] 44a1 ([2, 3], 1) [3, 1] 45a1 ([2], 1) [2, 1] 46a1 ([2], 1) [2, 1] 48a1 ([2], 1) [2, 2] 49a1 (0, 0) [2] 50a1 ([2, 3, 5], 1) [1, 3] 50b1 ([2, 3, 5], 1) [5, 1] 51a1 ([2, 3], 1) [3, 1] 52a1 ([2], 1) [1, 2] 53a1 ([2], 1) [1] 54a1 ([2, 3], 1) [1, 3] 54b1 ([2, 3], 1) [3, 1] 55a1 ([2], 1) [2, 2] 56a1 ([2], 1) [4, 1] 56b1 ([2], 1) [2, 1] 57a1 ([2], 1) [2, 1] 57c1 ([2, 5], 1) [10, 1] 57b1 ([2], 1) [2, 2] 58a1 ([2], 1) [2, 1] 58b1 ([2, 5], 1) [10, 1] 61a1 ([2], 1) [1] 62a1 ([2], 1) [4, 1] 63a1 ([2], 1) [2, 1] 64a1 (0, 0) [4] 65a1 ([2], 1) [1, 1] 66a1 ([2, 3], 1) [2, 3, 1] 66c1 ([2, 5], 25) [10, 5, 1] 66b1 ([2], 1) [4, 1, 1] 67a1 ([2], 1) [1] 69a1 ([2], 1) [2, 1] 70a1 ([2], 1) [4, 2, 1] 72a1 ([2], 1) [2, 4] 73a1 ([2], 1) [2] 75a1 ([2, 5], 1) [1, 1] 75c1 ([2, 5], 1) [5, 1] 75b1 ([2], 1) [1, 2] 76a1 ([2], 1) [1, 1] 77a1 ([2], 1) [2, 1] 77c1 ([2], 1) [1, 2] 77b1 ([2, 3], 1) [6, 1] 78a1 ([2], 1) [2, 1, 1] 79a1 ([2], 1) [1] 80a1 ([2], 1) [2, 2] 80b1 ([2, 3], 1) [1, 2] 82a1 ([2], 1) [2, 1] 83a1 ([2], 1) [1] 84a1 ([2, 3], 9) [3, 3, 2] 84b1 ([2], 1) [1, 1, 2] 85a1 ([2], 1) [2, 1] 88a1 ([2], 1) [4, 1] 89a1 ([2], 1) [1] 89b1 ([2], 1) [2] 90a1 ([2, 3], 1) [2, 2, 3] 90c1 ([2, 3], 1) [4, 4, 1] 90b1 ([2, 3], 1) [6, 2, 1] 91a1 ([2], 1) [1, 1] 91b1 ([2, 3], 1) [1, 1] 92a1 ([2, 3], 1) [3, 1] 92b1 ([2, 3], 9) [3, 1] 94a1 ([2], 1) [2, 1] 96a1 ([2], 1) [2, 2] 96b1 ([2], 1) [2, 2] 98a1 ([2, 3], 1) [2, 2] 99a1 ([2], 1) [2, 1] 99c1 ([2], 1) [2, 1] 99b1 ([2], 1) [4, 1] 99d1 ([2, 5], 1) [1, 1]```

Next we compute all curves (of any rank) in Cremona's book that have a prime pÕ5  that divides a Tamagawa number. For all other curves of rank Ô1  in Cremona's book, we are able to rule out the possibility that any other primes divide Sha using a calculation like above and Kato's theorem (and 3-descent).
 ```11a1 0 [5] 26b1 0 [7] 38b1 0 [5] 50b1 0 [5] 57c1 0 [2, 5] 58b1 0 [2, 5] 66c1 0 [2, 5] 66c1 0 [2, 5] 75c1 0 [5] 110a1 0 [5] 110a1 0 [5] 114c1 0 [2, 5] 118b1 0 [2, 5] 123a1 1 [5] 141a1 1 [7] 155a1 1 [5] 158c1 0 [2, 5] 170c1 0 [3, 7] 174a1 0 [3, 7] 174b1 0 [7] 174b1 0 [7] 182a1 0 [2, 5] 186b1 0 [5] 186b1 0 [5] 190a1 1 [2, 11] 203a1 0 [5] 214a1 1 [7] 238a1 1 [2, 7] 246b1 0 [5] 246b1 0 [5] 258f1 0 [2, 7] 258f1 0 [2, 7] 258c1 1 [2, 5] 262a1 1 [11] 264d1 0 [2, 7] 270b1 0 [3, 5] 274a1 1 [7] 280b1 1 [2, 3, 5] 285a1 1 [2, 5] 286b1 1 [2, 13] 286d1 0 [2, 5] 286d1 0 [2, 5] 302a1 1 [3, 5] 302c1 1 [5] 303a1 1 [2, 7] 309a1 1 [5] 318d1 1 [2, 11] 322d1 1 [2, 5] 325e1 0 [5] 326b1 1 [5] 330d1 0 [2, 7] 345c1 0 [2, 5] 346b1 1 [7] 348d1 1 [3, 7] 350f1 1 [2, 3, 11] 354f1 1 [2, 7] 354e1 0 [2, 11] 357d1 1 [2, 7] 362b1 1 [7] 364a1 1 [3, 5] 366g1 1 [2, 5] 366b1 0 [5] 366b1 0 [5] 366d1 0 [7] 378g1 0 [5] 381a1 1 [5] 395c1 0 [5] 406d1 0 [2, 5] 408d1 1 [2, 5] 414d1 1 [2, 5] 418b1 1 [2, 13] 426a1 0 [5] 426a1 0 [5] 426c1 0 [3, 5] 430b1 1 [5] 430d1 1 [3, 5] 430d1 1 [3, 5] 434d1 1 [2, 5] 442e1 0 [2, 11] 446b1 1 [2, 7] 458b1 1 [2, 5] 462e1 1 [2, 3, 7] 470f1 1 [2, 3, 7] 470c1 1 [2, 7] 474b1 1 [2, 5] 483a1 0 [5] 490g1 1 [2, 5] 490j1 0 [2, 5] 494d1 1 [2, 3, 13] 497a1 1 [5] 498b1 1 [2, 5] 506f1 1 [13] 506d1 1 [5] 522g1 0 [2, 11] 522i1 1 [2, 5] 522j1 1 [2, 13] 522m1 0 [2, 11] 528i1 0 [2, 5] 530c1 1 [2, 5] 537e1 0 [2, 5] 542a1 0 [2, 7] 542b1 1 [7] 546f1 0 [7] 546f1 0 [7] 546f1 0 [7] 546e1 0 [17] 550i1 1 [2, 3, 7] 550j1 1 [2, 11] 550m1 0 [11] 551c1 1 [2, 7] 555b1 0 [3, 5] 558f1 1 [2, 3, 5] 558g1 1 [2, 7] 560b1 0 [5] 560e1 1 [2, 5] 561b1 1 [2, 5] 570j1 0 [2, 7] 570l1 0 [2, 5] 570l1 0 [2, 5] 570l1 0 [2, 5] 570d1 0 [2, 5] 573b1 0 [5] 574g1 1 [11] 574i1 1 [3, 7] 574i1 1 [3, 7] 574j1 0 [5] 574j1 0 [5] 582c1 1 [2, 5] 585i1 1 [2, 7] 588d1 0 [5] 594h1 0 [5] 594d1 1 [2, 5] 595b1 0 [7] 598d1 1 [2, 17] 600e1 1 [2, 3, 7] 605a1 1 [3, 5] 605c1 1 [5] 606f1 0 [5] 606f1 0 [5] 606d1 0 [7] 608e1 1 [2, 5] 615b1 1 [2, 7] 616b1 0 [2, 5] 618f1 1 [7, 11] 618f1 1 [7, 11] 618e1 1 [5] 618d1 1 [2, 3, 5] 620b1 1 [2, 3, 5] 622a1 1 [7] 624i1 0 [2, 5] 624e1 0 [2, 5] 629d1 1 [5] 630g1 0 [2, 7] 642c1 1 [2, 13] 650k1 1 [2, 3, 7] 651a1 0 [2, 5] 658e1 1 [2, 11] 665a1 1 [5] 665d1 1 [2, 5] 665d1 1 [2, 5] 666g1 0 [2, 23] 666e1 1 [2, 13] 666d1 1 [2, 5] 670a1 1 [11] 670c1 1 [5] 670d1 1 [19] 672b1 1 [2, 3, 5] 674c1 1 [31] 678c1 1 [2, 7] 678d1 0 [2, 7] 678d1 0 [2, 7] 681e1 1 [2, 5] 682b1 1 [3, 19] 690g1 0 [2, 7] 690j1 0 [2, 5] 690e1 1 [2, 5] 696c1 1 [2, 5] 700d1 1 [2, 3, 5] 702j1 0 [11] 702k1 1 [3, 7] 702m1 1 [3, 19] 702l1 1 [2, 3, 5] 705b1 1 [3, 5] 705e1 1 [5] 706b1 1 [23] 706d1 1 [2, 5] 710c1 1 [2, 7] 710b1 1 [2, 17] 710d1 0 [2, 5] 710d1 0 [2, 5] 714c1 0 [5] 714e1 0 [7] 715b1 1 [3, 7] 726g1 1 [2, 3, 5] 726e1 1 [2, 5] 730i1 1 [7] 730j1 1 [3, 7] 734a1 0 [2, 5] 735f1 1 [2, 3, 7] 738f1 1 [2, 11] 738h1 0 [2, 7] 738e1 1 [2, 5] 741c1 0 [11] 741d1 0 [2, 5] 742g1 1 [2, 5] 742e1 1 [2, 5] 755f1 0 [13] 760b1 0 [2, 7] 762g1 0 [2, 3, 7] 762g1 0 [2, 3, 7] 762e1 1 [2, 3, 11] 762d1 1 [2, 5] 770c1 0 [2, 5] 774h1 0 [2, 7] 777g1 1 [2, 5] 777e1 1 [5] 780b1 0 [5] 782c1 0 [2, 7] 782e1 0 [2, 5] 786h1 1 [2, 7] 786j1 1 [3, 7] 786m1 0 [3, 5] 786m1 0 [3, 5] 786l1 1 [5, 7] 786l1 1 [5, 7] 794c1 1 [5] 795c1 0 [3, 5] 798g1 1 [2, 3, 5] 798h1 1 [2, 3, 7] 798c1 1 [2, 5] 798d1 1 [2, 5] 804d1 1 [3, 7] 806f1 0 [2, 5] 806f1 0 [2, 5] 806c1 1 [2, 5] 806d1 1 [2, 3, 11] 810g1 0 [3, 5] 814b1 1 [5] 816i1 1 [2, 11] 817b1 1 [2, 5] 822d1 1 [2, 5] 830c1 1 [2, 5] 831a1 1 [2, 5] 834f1 1 [2, 7] 834g1 1 [2, 5] 834g1 1 [2, 5] 834a1 0 [2, 7] 840d1 0 [2, 5] 842b1 1 [13] 850l1 1 [2, 7] 850d1 1 [2, 7] 854d1 1 [2, 3, 7] 858f1 1 [2, 5, 11] 858f1 1 [2, 5, 11] 858k1 0 [2, 7] 858k1 0 [2, 7] 858k1 0 [2, 7] 858l1 0 [2, 7] 861c1 1 [5, 7] 861c1 1 [5, 7] 861b1 1 [17] 861d1 1 [5] 862e1 1 [2, 5] 870f1 1 [2, 5, 7] 870f1 1 [2, 5, 7] 870i1 0 [2, 5] 870i1 0 [2, 5] 870i1 0 [2, 5] 874f1 0 [3, 7] 874e1 1 [5] 874e1 1 [5] 874d1 1 [5] 876b1 1 [3, 5] 880g1 1 [2, 5] 882h1 1 [2, 3, 5] 882j1 0 [2, 5] 885d1 1 [5] 885d1 1 [5] 886e1 1 [5] 886d1 1 [2, 19] 890f1 1 [13] 890g1 1 [5] 890g1 1 [5] 894f1 1 [5] 894g1 1 [7, 11] 894g1 1 [7, 11] 894c1 0 [3, 5] 894e1 1 [2, 23] 897e1 1 [2, 5] 897d1 1 [2, 3, 5] 901e1 1 [3, 5] 905b1 0 [5] 906h1 1 [5, 11] 906h1 1 [5, 11] 906e1 0 [5] 910f1 1 [2, 5, 11] 910f1 1 [2, 5, 11] 910g1 1 [2, 5] 910h1 1 [2, 3, 17] 910k1 1 [2, 5, 7] 910k1 1 [2, 5, 7] 912h1 1 [2, 5] 915a1 0 [7] 918h1 1 [3, 11] 918j1 1 [2, 3, 5] 920a1 1 [2, 3, 5] 924f1 0 [5] 924h1 1 [3, 5] 924b1 1 [3, 5] 924e1 1 [3, 5] 930f1 0 [11] 930h1 1 [2, 3, 5] 930d1 1 [2, 7] 933b1 1 [11] 934b1 0 [3, 5] 938b1 1 [2, 5] 939c1 1 [5] 942c1 1 [2, 5] 946c1 0 [2, 5] 954i1 1 [2, 5] 954h1 1 [2, 7] 954j1 1 [2, 17] 966h1 0 [5] 966b1 0 [5] 974h1 1 [3, 5] 975i1 1 [2, 3, 7] 975j1 1 [2, 5] 978f1 1 [2, 11] 978g1 1 [2, 7] 986e1 1 [2, 5, 7] 986e1 1 [2, 5, 7] 987e1 1 [2, 3, 5] 987d1 0 [7] 988b1 1 [3, 13] 990l1 0 [7] 996b1 1 [3, 13]``` ```11a1 0 [5] 26b1 0 [7] 38b1 0 [5] 50b1 0 [5] 57c1 0 [2, 5] 58b1 0 [2, 5] 66c1 0 [2, 5] 66c1 0 [2, 5] 75c1 0 [5] 110a1 0 [5] 110a1 0 [5] 114c1 0 [2, 5] 118b1 0 [2, 5] 123a1 1 [5] 141a1 1 [7] 155a1 1 [5] 158c1 0 [2, 5] 170c1 0 [3, 7] 174a1 0 [3, 7] 174b1 0 [7] 174b1 0 [7] 182a1 0 [2, 5] 186b1 0 [5] 186b1 0 [5] 190a1 1 [2, 11] 203a1 0 [5] 214a1 1 [7] 238a1 1 [2, 7] 246b1 0 [5] 246b1 0 [5] 258f1 0 [2, 7] 258f1 0 [2, 7] 258c1 1 [2, 5] 262a1 1 [11] 264d1 0 [2, 7] 270b1 0 [3, 5] 274a1 1 [7] 280b1 1 [2, 3, 5] 285a1 1 [2, 5] 286b1 1 [2, 13] 286d1 0 [2, 5] 286d1 0 [2, 5] 302a1 1 [3, 5] 302c1 1 [5] 303a1 1 [2, 7] 309a1 1 [5] 318d1 1 [2, 11] 322d1 1 [2, 5] 325e1 0 [5] 326b1 1 [5] 330d1 0 [2, 7] 345c1 0 [2, 5] 346b1 1 [7] 348d1 1 [3, 7] 350f1 1 [2, 3, 11] 354f1 1 [2, 7] 354e1 0 [2, 11] 357d1 1 [2, 7] 362b1 1 [7] 364a1 1 [3, 5] 366g1 1 [2, 5] 366b1 0 [5] 366b1 0 [5] 366d1 0 [7] 378g1 0 [5] 381a1 1 [5] 395c1 0 [5] 406d1 0 [2, 5] 408d1 1 [2, 5] 414d1 1 [2, 5] 418b1 1 [2, 13] 426a1 0 [5] 426a1 0 [5] 426c1 0 [3, 5] 430b1 1 [5] 430d1 1 [3, 5] 430d1 1 [3, 5] 434d1 1 [2, 5] 442e1 0 [2, 11] 446b1 1 [2, 7] 458b1 1 [2, 5] 462e1 1 [2, 3, 7] 470f1 1 [2, 3, 7] 470c1 1 [2, 7] 474b1 1 [2, 5] 483a1 0 [5] 490g1 1 [2, 5] 490j1 0 [2, 5] 494d1 1 [2, 3, 13] 497a1 1 [5] 498b1 1 [2, 5] 506f1 1 [13] 506d1 1 [5] 522g1 0 [2, 11] 522i1 1 [2, 5] 522j1 1 [2, 13] 522m1 0 [2, 11] 528i1 0 [2, 5] 530c1 1 [2, 5] 537e1 0 [2, 5] 542a1 0 [2, 7] 542b1 1 [7] 546f1 0 [7] 546f1 0 [7] 546f1 0 [7] 546e1 0 [17] 550i1 1 [2, 3, 7] 550j1 1 [2, 11] 550m1 0 [11] 551c1 1 [2, 7] 555b1 0 [3, 5] 558f1 1 [2, 3, 5] 558g1 1 [2, 7] 560b1 0 [5] 560e1 1 [2, 5] 561b1 1 [2, 5] 570j1 0 [2, 7] 570l1 0 [2, 5] 570l1 0 [2, 5] 570l1 0 [2, 5] 570d1 0 [2, 5] 573b1 0 [5] 574g1 1 [11] 574i1 1 [3, 7] 574i1 1 [3, 7] 574j1 0 [5] 574j1 0 [5] 582c1 1 [2, 5] 585i1 1 [2, 7] 588d1 0 [5] 594h1 0 [5] 594d1 1 [2, 5] 595b1 0 [7] 598d1 1 [2, 17] 600e1 1 [2, 3, 7] 605a1 1 [3, 5] 605c1 1 [5] 606f1 0 [5] 606f1 0 [5] 606d1 0 [7] 608e1 1 [2, 5] 615b1 1 [2, 7] 616b1 0 [2, 5] 618f1 1 [7, 11] 618f1 1 [7, 11] 618e1 1 [5] 618d1 1 [2, 3, 5] 620b1 1 [2, 3, 5] 622a1 1 [7] 624i1 0 [2, 5] 624e1 0 [2, 5] 629d1 1 [5] 630g1 0 [2, 7] 642c1 1 [2, 13] 650k1 1 [2, 3, 7] 651a1 0 [2, 5] 658e1 1 [2, 11] 665a1 1 [5] 665d1 1 [2, 5] 665d1 1 [2, 5] 666g1 0 [2, 23] 666e1 1 [2, 13] 666d1 1 [2, 5] 670a1 1 [11] 670c1 1 [5] 670d1 1 [19] 672b1 1 [2, 3, 5] 674c1 1 [31] 678c1 1 [2, 7] 678d1 0 [2, 7] 678d1 0 [2, 7] 681e1 1 [2, 5] 682b1 1 [3, 19] 690g1 0 [2, 7] 690j1 0 [2, 5] 690e1 1 [2, 5] 696c1 1 [2, 5] 700d1 1 [2, 3, 5] 702j1 0 [11] 702k1 1 [3, 7] 702m1 1 [3, 19] 702l1 1 [2, 3, 5] 705b1 1 [3, 5] 705e1 1 [5] 706b1 1 [23] 706d1 1 [2, 5] 710c1 1 [2, 7] 710b1 1 [2, 17] 710d1 0 [2, 5] 710d1 0 [2, 5] 714c1 0 [5] 714e1 0 [7] 715b1 1 [3, 7] 726g1 1 [2, 3, 5] 726e1 1 [2, 5] 730i1 1 [7] 730j1 1 [3, 7] 734a1 0 [2, 5] 735f1 1 [2, 3, 7] 738f1 1 [2, 11] 738h1 0 [2, 7] 738e1 1 [2, 5] 741c1 0 [11] 741d1 0 [2, 5] 742g1 1 [2, 5] 742e1 1 [2, 5] 755f1 0 [13] 760b1 0 [2, 7] 762g1 0 [2, 3, 7] 762g1 0 [2, 3, 7] 762e1 1 [2, 3, 11] 762d1 1 [2, 5] 770c1 0 [2, 5] 774h1 0 [2, 7] 777g1 1 [2, 5] 777e1 1 [5] 780b1 0 [5] 782c1 0 [2, 7] 782e1 0 [2, 5] 786h1 1 [2, 7] 786j1 1 [3, 7] 786m1 0 [3, 5] 786m1 0 [3, 5] 786l1 1 [5, 7] 786l1 1 [5, 7] 794c1 1 [5] 795c1 0 [3, 5] 798g1 1 [2, 3, 5] 798h1 1 [2, 3, 7] 798c1 1 [2, 5] 798d1 1 [2, 5] 804d1 1 [3, 7] 806f1 0 [2, 5] 806f1 0 [2, 5] 806c1 1 [2, 5] 806d1 1 [2, 3, 11] 810g1 0 [3, 5] 814b1 1 [5] 816i1 1 [2, 11] 817b1 1 [2, 5] 822d1 1 [2, 5] 830c1 1 [2, 5] 831a1 1 [2, 5] 834f1 1 [2, 7] 834g1 1 [2, 5] 834g1 1 [2, 5] 834a1 0 [2, 7] 840d1 0 [2, 5] 842b1 1 [13] 850l1 1 [2, 7] 850d1 1 [2, 7] 854d1 1 [2, 3, 7] 858f1 1 [2, 5, 11] 858f1 1 [2, 5, 11] 858k1 0 [2, 7] 858k1 0 [2, 7] 858k1 0 [2, 7] 858l1 0 [2, 7] 861c1 1 [5, 7] 861c1 1 [5, 7] 861b1 1 [17] 861d1 1 [5] 862e1 1 [2, 5] 870f1 1 [2, 5, 7] 870f1 1 [2, 5, 7] 870i1 0 [2, 5] 870i1 0 [2, 5] 870i1 0 [2, 5] 874f1 0 [3, 7] 874e1 1 [5] 874e1 1 [5] 874d1 1 [5] 876b1 1 [3, 5] 880g1 1 [2, 5] 882h1 1 [2, 3, 5] 882j1 0 [2, 5] 885d1 1 [5] 885d1 1 [5] 886e1 1 [5] 886d1 1 [2, 19] 890f1 1 [13] 890g1 1 [5] 890g1 1 [5] 894f1 1 [5] 894g1 1 [7, 11] 894g1 1 [7, 11] 894c1 0 [3, 5] 894e1 1 [2, 23] 897e1 1 [2, 5] 897d1 1 [2, 3, 5] 901e1 1 [3, 5] 905b1 0 [5] 906h1 1 [5, 11] 906h1 1 [5, 11] 906e1 0 [5] 910f1 1 [2, 5, 11] 910f1 1 [2, 5, 11] 910g1 1 [2, 5] 910h1 1 [2, 3, 17] 910k1 1 [2, 5, 7] 910k1 1 [2, 5, 7] 912h1 1 [2, 5] 915a1 0 [7] 918h1 1 [3, 11] 918j1 1 [2, 3, 5] 920a1 1 [2, 3, 5] 924f1 0 [5] 924h1 1 [3, 5] 924b1 1 [3, 5] 924e1 1 [3, 5] 930f1 0 [11] 930h1 1 [2, 3, 5] 930d1 1 [2, 7] 933b1 1 [11] 934b1 0 [3, 5] 938b1 1 [2, 5] 939c1 1 [5] 942c1 1 [2, 5] 946c1 0 [2, 5] 954i1 1 [2, 5] 954h1 1 [2, 7] 954j1 1 [2, 17] 966h1 0 [5] 966b1 0 [5] 974h1 1 [3, 5] 975i1 1 [2, 3, 7] 975j1 1 [2, 5] 978f1 1 [2, 11] 978g1 1 [2, 7] 986e1 1 [2, 5, 7] 986e1 1 [2, 5, 7] 987e1 1 [2, 3, 5] 987d1 0 [7] 988b1 1 [3, 13] 990l1 0 [7] 996b1 1 [3, 13]```

## Another Example: 894e1

Another difficult example is the curve 894e1.
• It has rank 1, so Kato doesn't apply.

• Refined Kolyvagin shows that #Sha(E)  is either trivial or of order 232 . But we'll probably never directly compute the 23 -Selmer group.

 `1` `1`
 `(0, 0)` `(0, 0)`
 `[-47, -95]` `[-47, -95]`
 `[-0.00018561213 ... 0.00018561213]` `[-0.00018561213 ... 0.00018561213]`
 `[2115.9843 ... 2116.0118]` `[2115.9843 ... 2116.0118]`
 `([2, 23], 529)` `([2, 23], 529)`
 `True` `True`
 `23^2` `23^2`

## NEXT: Iwasawa theory and p -adic Analogues of the BSD Conjecture

 New Delete  Worksheets Name: Password: X    X [-] Saved Objects [-] Variables E (sage.schemes.elliptic_curves.ell_rational_field.EllipticCurve_rational_field)e (sage.schemes.elliptic_curves.ell_rational_field.EllipticCurve_rational_field) [-] Attached Files /Users/was/.sage/init.sage