Hi All,12This email concerns where the invisible examples of Sha in3[Cremona-Mazur] become visible. I've found a method to get further4information, and analyzed each of the cases in Table 1 of5[Cremona-Mazur]. I've talked to all of you about this problem at one6point or another, so I thought you might find these computations7interesting. The discussion of each case follows. The most8interesting examples are 2849A and 5389A.9102849A: Barry and Adam found the first known example of an _invisible_11Shafarevich-Tate group. This was Sha(E), where E is the elliptic12curve 2849A, which is given by the minimal Weierstrass equation1314y^2 + x*y + y = x^3 + x^2 - 53484*x - 4843180.1516I found an elliptic curve F of conductor 8547=2839*3 such that f_E =17f_F (mod 3), where f_E and f_F are the newforms attached to E and F.18Furthermore the Mordell-Weil group of F has rank TWO. The equation of19F is2021y^2 + x*y + y = x^3 + x^2 - 154*x - 4782223Thus maybe Sha(E) becomes visible at level 8547. (This isn't24completely obvious to me because the geometric, but not arithmetic,25component group of E at 3 has order necessarily divisible by 3.)26274343B: Again, according to Table 1 of [Cremona-Mazur], Sha(E) has28order 9, but the modular degree prevents Sha(E) from being visible.29At level 5*4343 there is an elliptic curve F of rank 1 that is30congruent to E. Its equation is3132y^2 - x*y - y = x^3 - x^2 + 78*x - 256333435HOW I FOUND THESE CURVES: Let me explain the naive observation that36allowed me find these curves of higher level, even though their37conductors are out of the range of Cremona's tables. The elliptic38curves labeled "NONE" in table 1 of [Cremona-Mazur] have unusually39large height, as compared to their conductors. However, the40explanatory factors often have unusually small height. So I just made41a table of all elliptic curves [*, *, *, a4, a6] with * in {-1,0,1}42and a4, a6 bounded by 1000 and of conductor < 50,000.43443364C, 4229A, 5073d: I tried this method on the curves 3364C, 4229A,455073D each of which is labeled "none" (as opposed to NONE) in46Cremona-Mazur. In none of these 3 cases was I able to find an47explanatory factor at higher level.48494194N: The curve 4914N is labeled "none" and has the remark "E has50rational 3-torsion", even though there are invisible elements in Sha51of order 3. Here I was able to find a congruent curve at level 24570.52The congruent curve is5354F: y^2 - x*y = x^3 - x^2 - 15*x - 75.5556It has completely trivial Mordell-Weil group: F(Q) = {0}; there isn't57even 3-torsion, so maybe Sha is not visible in E+F.58595054C: The curve 5054C is labeled "none" and has invisible elements of60Sha of order 3. I found a congruent curve of level 25270 of rank 1.61The equation of the congruent curve is6263F: y^2 - x*y = x^3 + x^2 - 178*x + 882.64655389A: The last curve labeled "NONE" in the table is curve 5389A. I66do not know the equation for this curve, because the height is not67small, John has not published it, and I have not bothered to compute68it. However I found the eigenvalues a_p using modular symbols; they69are7071[1, -2, 2, -2, 6, 2, 1, -4, 6, ...].7273We can raise the level at 3, since (-2)^2 = (3+1)^2 (mod 3). However,74my table of curves of small height does not have any of conductor753*5389. However, (-2)^2 = (7+1)^2 (mod 3), so we can instead add 776into the level. My table does have a curve of conductor 37723, and77*luckily* it is congruent mod 3 to 5389A! It's Weierstrass equations78is7980y^2 - y = x^3 + x^2 + 34*x - 248.8182According to Cremona's mwrank program, this curve at level 37723 has83rank 2.8485Please share your comments or suggestions with me.8687Best regards,88William899091---------------------------------------92Loic asked:9394> Thanks for the news. Is it known that the example in question does95> not become visible in J_1(2849)?9697To the best of my knowledge this still isn't known. I haven't asked98Mazur in a while, though.99100I just tried a computation which *might* show that the element does not101become visible in J_1(2849). However, I haven't thought hard enough102about it to decide whether or not the implication is justified. Here's103what I did. Let M_2(2849,F_3)(eps) denote the space of modular symbols104computed modulo 3 using the relations from your SLNM article. There105are 8 Dirichlet character eps:(Z/2849Z)^* ---> F_3^*. Four of them are106odd, and for these the corresponding space of modular symbols is107trivial. The other four are even, and the corresponding spaces of108modular symbols are F_3-vector spaces of dimension just over 600.109I searched in each of these spaces and found *exactly one* eigenvector110with the same Hecke eigenvalues as that of the elliptic curve 2849A.111This one form is the mod-3 reduction reduction of eigenvector112corresponding to 2849A.113114What do you think?115116Regards,117William118119120> EulerPhi(2849);1212160122> G:=DirichletGroup(2849,GF(3));123> e:=Elements(G);124> SetVerbose("ModularForm",2);125> E:=EllipticCurve(CremonaDatabase(),"2849A");126> E;127Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 - 53484*x - 4843180 over Rational Field128> factor(2849);129[ <7, 1>, <11, 1>, <37, 1> ]130> ellap(E,2);131-1132> ellap(E,3);1332134> ellap(E,5);135-2136> R<x>:=PolynomialRing(GF(3));137> I:=[<2,x-ellap(E,2)>,<3,x-ellap(E,3)>,<5,x-ellap(E,5)>];138> time M := [MS(e[i],2,+1) : i in [1..#e]];139Time: 63.620140> Mnz:=[m : m in M | Dimension(m) ne 0];141> Mnz;142[143Full Modular symbols space of level 2849, weight 2, and dimension 308,144Full Modular symbols space of level 2849, weight 2, character $.1*$.2, and dimension 304,145Full Modular symbols space of level 2849, weight 2, character $.3, and dimension 308,146Full Modular symbols space of level 2849, weight 2, character $.1*$.2*$.3, and dimension 304147]148> time K1:=Kernel(I,Mnz[1]);149Time: 1.700150> K1;151Modular symbols space of level 2849, weight 2, and dimension 1152> SetVerbose("ModularForm",0);153> time K2:=Kernel(I,Mnz[2]);154Time: 3.580155> K2;156Modular symbols space of level 2849, weight 2, character $.1*$.2, and dimension 2157> DirichletCharacter(Mnz[2]);158$.1*$.2159> qEigenform(K1,40);160q + 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)161> DualHeckeOperator(K2,13);DualHeckeOperator(K2,13);162[1 0]163[0 1]164> DualHeckeOperator(K2,31);165[1 1]166[0 1]167> qEigenform(K1,40);168q + 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)169> DualHeckeOperator(K2,31);170[0 0]171[0 0]172> I:=[<p,x-ellap(E,p)> : p in [2,3,5,13,17,19,23,29,31]];173> time K1:=Kernel(I,Mnz[1]);174Time: 1.369175> K1;176Modular symbols space of level 2849, weight 2, and dimension 1177> time K2:=Kernel(I,Mnz[2]);178Time: 1.679179> K2;180Modular symbols space of level 2849, weight 2, character $.1*$.2, and dimension 0181> time K3:=Kernel(I,Mnz[3]);182Time: 3.890183> K3;184Modular symbols space of level 2849, weight 2, character $.3, and dimension 0185> time K4:=Kernel(I,Mnz[4]);186Time: 2.800187> K4;188Modular symbols space of level 2849, weight 2, character $.1*$.2*$.3, and dimension 0189> #Mnz;1904191> Mnz[1];192Full Modular symbols space of level 2849, weight 2, and dimension 308193>194