What is the meaning of mod in quadray of Pari software

Question

I need a polynomial that defines the narrow Hilbert class field of the real quadratic field x^2-505. How quadray could do this? quadray(x^2-505,-1)? What is the meaning of Mod in output?

x^4 + Mod(-19*y - 207, y^2 - y - 126)*x^3 + Mod(305*y + 3277, y^2 - y - 126)*x^2 + Mod(-1523*y - 16351, y^2 - y - 126)*x + Mod(21732 - y - 126)

Answer 1

Mod(y, y^2 - y - 126) means "a root of y^2 - y - 126". What you obtained is an equation with coefficients in the quadratic field Q[y]/(y^2 - y - 126) which is isomorphic to Q(sqrt(505)) since the discriinant of y^2 - y - 126 is 505, i.e., a relative extension of degree 4 over that quadratic field.

However, this is not the answer you're looking for: quadray(505,-1) just computes the ordinary Hilbert class field, not the narrow one. (The -1 is the conductor ideal and (-1) = (1).) Indeed, bnfnarrow(bnfinit(y^2-505)) tells you that you're looking for a degree 8 extension, not 4 ! The quadray function is not suitable, use

bnrclassfield(bnrinit(bnfinit(y^2-505), [1,[1,1]]))

which yields

x^8 + 2*x^7 + (-y + 2)*x^6 + (3*y + 77)*x^5 + (55/2*y + 1317/2)*x^4 + (16*y + 380)*x^3 + (-255/2*y - 5685/2)*x^2 + (95*y + 2125)*x + (650*y + 14625)

This is again a relative extension of degree 8 over Q(sqrt(505)), except this time y = sqrt(505), since I specified the base by the explicit polynomial y^2 - 505.

What [1, [1,1]] means is : a modulus (in the sense of class field theory), where we allow ramification at no finite place (the initial 1), but at both places at infinity (the [1,1]).