/* Base fields: orbit 48g: Q(a) orbit 48f: Q(a) orbit 48e: Q(a) orbit 48abcd: Q(a,r3,b4) SIC fields: orbit 48g (ray class field): Q(a,r1,r2,t,b1,b2,b3,i) (degree 512) orbit 48f: Q(a,r1,r2,t,b1,b2,b3,b6,i) (degree 1536) orbit 48e: Q(a,r1,r2,r3,t,b1,b2,b3,b4,b5,i) (degree 4096) orbit 48abcd: Q(a,r1,r2,r3,b1,b2,b3,b4,b5,b6,i) (degree 12288) */ P0:=PolynomialRing(RationalField()); Fa:=NumberField(x0^2-5); Pa:=PolynomialRing(Fa); Fr1:=NumberField(xa^2-2); Pr1:=PolynomialRing(Fr1); Fr2:=NumberField(xr1^2-3); Pr2:=PolynomialRing(Fr2); Fr3:=NumberField(xr2^2-105); Pr3:=PolynomialRing(Fr3); Ft:=NumberField(32*xr3^4-32*xr3^2+(4-r1-r1*r2)); Pt:=PolynomialRing(Ft); Fb4:=NumberField(xt^2-(2*r3+42)); Pb4:=PolynomialRing(Fb4); // Irreducibility of the remaining defining polynomials was checked indirectly, using the argument described below. Fb2:=NumberField(xb4^2-(6+r1*r2): Check:=false); Pb2:=PolynomialRing(Fb2); Fb3:=NumberField(xb2^2-(6-2*r1*r2+(r2-r1)*b2): Check:=false); Pb3:=PolynomialRing(Fb3); Fb5:=NumberField(xb3^2+((3*a-5)*r3+(-5*a-5))*b4+(2*a+10)*r3+210*a-630: Check:=false); Pb5:=PolynomialRing(Fb5); Fb6:=NumberField(xb5^3 - 12*xb5 - 14: Check:=false); Pb6:=PolynomialRing(Fb6); Fb1:=NumberField(xb6^2+(a-1): Check:=false); Pb1:=PolynomialRing(Fb1); F:=NumberField(xb1^2+1: Check:=false); P:=PolynomialRing(F); tau:=((-4*r1*r2-4*r1)*t^3+((3*r1+1)*r2+(3*r1+2))*t)*i-t; /* CHECKING IRREDUCIBILITY OF THE DEFINING POLYNOMIALS OF Fb2, Fb3, Fb5, Fb6, Fb1, F. A direct calculation would take to long. So we rely on the following indirect approach. Define Fi = Q(a,r1,r2,t,b1,b2,b3,i) is the degree 512 ray class field Fb5= Q(a,r3,b4,b5) is degree 16 Fb6 = Q(b6) is degree 3 We first show that Fb5 cap Fi = Q(a). Let G = Fb5 cap Fi. xi^2-105 is irreducible over Fi (1000 seconds). So G does not contain r3. The defining polynomials of the subfields of Fb5 are [ x0^16 - 5040*x0^14 + 9004800*x0^12 - 7010304000*x0^10 + 2552954880000*x0^8 - 421960089600000*x0^6 + 28554559488000000*x0^4 - 490167336960000000*x0^2 + 739875225600000000, x0^2 - 19783680000*x0 + 739875225600000000, x0^2 - 276971520000*x0 + 739875225600000000, x0^2 - 70533120000*x0 + 739875225600000000, x0^4 - 403200*x0^3 + 34406400000*x0^2 - 809238528000000*x0 + 739875225600000000, x0^4 - 940800*x0^3 + 34406400000*x0^2 - 346816512000000*x0 + 739875225600000000, x0^4 - 1344000*x0^3 + 367288320000*x0^2 - 19652935680000000*x0 + 739875225600000000, x0^4 - 940800*x0^3 + 110100480000*x0^2 - 3121348608000000*x0 + 739875225600000000, x0^4 - 532564*x0^3 + 72059216886*x0^2 - 306894451300084*x0 + 277843033991975761, x0^8 + 5040*x0^7 + 9004800*x0^6 + 7010304000*x0^5 + 2552954880000*x0^4 + 421960089600000*x0^3 + 28554559488000000*x0^2 + 490167336960000000*x0 + 739875225600000000 ] // G is a subfield containing a but not r3. We find that the only such subfield is the third on the above list, which is just Q(a). We have thus established that Fb5 meet Fi = G = Q(a) which means Fb5 join Fi = Q(a,r1,r2,r3,t,b1,b2,b3,b4,b5,i) is degree 2^11 = 4096. The degree of (Fb5 join Fi) meet Fb6 must divide both 4096 and 3, which means the degree is 1. It follows that each of the polynomials used to build the tower in d48BField.mgm is irreducible over the field before. */