Hi All, This email concerns where the invisible examples of Sha in [Cremona-Mazur] become visible. I've found a method to get further information, and analyzed each of the cases in Table 1 of [Cremona-Mazur]. I've talked to all of you about this problem at one point or another, so I thought you might find these computations interesting. The discussion of each case follows. The most interesting examples are 2849A and 5389A. 2849A: Barry and Adam found the first known example of an _invisible_ Shafarevich-Tate group. This was Sha(E), where E is the elliptic curve 2849A, which is given by the minimal Weierstrass equation y^2 + x*y + y = x^3 + x^2 - 53484*x - 4843180. I found an elliptic curve F of conductor 8547=2839*3 such that f_E = f_F (mod 3), where f_E and f_F are the newforms attached to E and F. Furthermore the Mordell-Weil group of F has rank TWO. The equation of F is y^2 + x*y + y = x^3 + x^2 - 154*x - 478 Thus maybe Sha(E) becomes visible at level 8547. (This isn't completely obvious to me because the geometric, but not arithmetic, component group of E at 3 has order necessarily divisible by 3.) 4343B: Again, according to Table 1 of [Cremona-Mazur], Sha(E) has order 9, but the modular degree prevents Sha(E) from being visible. At level 5*4343 there is an elliptic curve F of rank 1 that is congruent to E. Its equation is y^2 - x*y - y = x^3 - x^2 + 78*x - 256 HOW I FOUND THESE CURVES: Let me explain the naive observation that allowed me find these curves of higher level, even though their conductors are out of the range of Cremona's tables. The elliptic curves labeled "NONE" in table 1 of [Cremona-Mazur] have unusually large height, as compared to their conductors. However, the explanatory factors often have unusually small height. So I just made a table of all elliptic curves [*, *, *, a4, a6] with * in {-1,0,1} and a4, a6 bounded by 1000 and of conductor < 50,000. 3364C, 4229A, 5073d: I tried this method on the curves 3364C, 4229A, 5073D each of which is labeled "none" (as opposed to NONE) in Cremona-Mazur. In none of these 3 cases was I able to find an explanatory factor at higher level. 4194N: The curve 4914N is labeled "none" and has the remark "E has rational 3-torsion", even though there are invisible elements in Sha of order 3. Here I was able to find a congruent curve at level 24570. The congruent curve is F: y^2 - x*y = x^3 - x^2 - 15*x - 75. It has completely trivial Mordell-Weil group: F(Q) = {0}; there isn't even 3-torsion, so maybe Sha is not visible in E+F. 5054C: The curve 5054C is labeled "none" and has invisible elements of Sha of order 3. I found a congruent curve of level 25270 of rank 1. The equation of the congruent curve is F: y^2 - x*y = x^3 + x^2 - 178*x + 882. 5389A: The last curve labeled "NONE" in the table is curve 5389A. I do not know the equation for this curve, because the height is not small, John has not published it, and I have not bothered to compute it. However I found the eigenvalues a_p using modular symbols; they are [1, -2, 2, -2, 6, 2, 1, -4, 6, ...]. We can raise the level at 3, since (-2)^2 = (3+1)^2 (mod 3). However, my table of curves of small height does not have any of conductor 3*5389. However, (-2)^2 = (7+1)^2 (mod 3), so we can instead add 7 into the level. My table does have a curve of conductor 37723, and *luckily* it is congruent mod 3 to 5389A! It's Weierstrass equations is y^2 - y = x^3 + x^2 + 34*x - 248. According to Cremona's mwrank program, this curve at level 37723 has rank 2. Please share your comments or suggestions with me. Best regards, William --------------------------------------- Loic asked: > Thanks for the news. Is it known that the example in question does > not become visible in J_1(2849)? To the best of my knowledge this still isn't known. I haven't asked Mazur in a while, though. I just tried a computation which *might* show that the element does not become visible in J_1(2849). However, I haven't thought hard enough about it to decide whether or not the implication is justified. Here's what I did. Let M_2(2849,F_3)(eps) denote the space of modular symbols computed modulo 3 using the relations from your SLNM article. There are 8 Dirichlet character eps:(Z/2849Z)^* ---> F_3^*. Four of them are odd, and for these the corresponding space of modular symbols is trivial. The other four are even, and the corresponding spaces of modular symbols are F_3-vector spaces of dimension just over 600. I searched in each of these spaces and found *exactly one* eigenvector with the same Hecke eigenvalues as that of the elliptic curve 2849A. This one form is the mod-3 reduction reduction of eigenvector corresponding to 2849A. What do you think? Regards, William > EulerPhi(2849); 2160 > G:=DirichletGroup(2849,GF(3)); > e:=Elements(G); > SetVerbose("ModularForm",2); > E:=EllipticCurve(CremonaDatabase(),"2849A"); > E; Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 - 53484*x - 4843180 over Rational Field > factor(2849); [ <7, 1>, <11, 1>, <37, 1> ] > ellap(E,2); -1 > ellap(E,3); 2 > ellap(E,5); -2 > R:=PolynomialRing(GF(3)); > I:=[<2,x-ellap(E,2)>,<3,x-ellap(E,3)>,<5,x-ellap(E,5)>]; > time M := [MS(e[i],2,+1) : i in [1..#e]]; Time: 63.620 > Mnz:=[m : m in M | Dimension(m) ne 0]; > Mnz; [ Full Modular symbols space of level 2849, weight 2, and dimension 308, Full Modular symbols space of level 2849, weight 2, character $.1*$.2, and dimension 304, Full Modular symbols space of level 2849, weight 2, character $.3, and dimension 308, Full Modular symbols space of level 2849, weight 2, character $.1*$.2*$.3, and dimension 304 ] > time K1:=Kernel(I,Mnz[1]); Time: 1.700 > K1; Modular symbols space of level 2849, weight 2, and dimension 1 > SetVerbose("ModularForm",0); > time K2:=Kernel(I,Mnz[2]); Time: 3.580 > K2; Modular symbols space of level 2849, weight 2, character $.1*$.2, and dimension 2 > DirichletCharacter(Mnz[2]); $.1*$.2 > qEigenform(K1,40); q + 2*q^2 + 2*q^3 + 2*q^4 + q^5 + q^6 + q^7 + q^9 + 2*q^10 + 2*q^11 + q^12 + q^13 + 2*q^14 + 2*q^15 + 2*q^16 + q^17 + 2*q^18 + 2*q^20 + 2*q^21 + q^22 + 2*q^25 + 2*q^26 + 2*q^27 + 2*q^28 + q^30 + q^31 + q^32 + q^33 + 2*q^34 + q^35 + 2*q^36 + O(q^37) > DualHeckeOperator(K2,13);DualHeckeOperator(K2,13); [1 0] [0 1] > DualHeckeOperator(K2,31); [1 1] [0 1] > qEigenform(K1,40); q + 2*q^2 + 2*q^3 + 2*q^4 + q^5 + q^6 + q^7 + q^9 + 2*q^10 + 2*q^11 + q^12 + q^13 + 2*q^14 + 2*q^15 + 2*q^16 + q^17 + 2*q^18 + 2*q^20 + 2*q^21 + q^22 + 2*q^25 + 2*q^26 + 2*q^27 + 2*q^28 + q^30 + q^31 + q^32 + q^33 + 2*q^34 + q^35 + 2*q^36 + 2*q^37 + 2*q^39 + O(q^40) > DualHeckeOperator(K2,31); [0 0] [0 0] > I:=[ : p in [2,3,5,13,17,19,23,29,31]]; > time K1:=Kernel(I,Mnz[1]); Time: 1.369 > K1; Modular symbols space of level 2849, weight 2, and dimension 1 > time K2:=Kernel(I,Mnz[2]); Time: 1.679 > K2; Modular symbols space of level 2849, weight 2, character $.1*$.2, and dimension 0 > time K3:=Kernel(I,Mnz[3]); Time: 3.890 > K3; Modular symbols space of level 2849, weight 2, character $.3, and dimension 0 > time K4:=Kernel(I,Mnz[4]); Time: 2.800 > K4; Modular symbols space of level 2849, weight 2, character $.1*$.2*$.3, and dimension 0 > #Mnz; 4 > Mnz[1]; Full Modular symbols space of level 2849, weight 2, and dimension 308 >