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From: David Bernier on 5 Aug 2010 14:50
According to a paper by Muzaffar and Williams,
< http://www.mathstat.carleton.ca/~williams/papers/pdf/269.pdf >

Weber's three functions f(z), f_1(z) and f_3(z) are determined
in terms of the Dedekind eta function, which is holomorphic
in the upper half of the complex plane.

From
http://www.numbertheory.org/classnos/

we have for a given class number for the ring of integers in
a quadratic imaginary field extension of Q either:

- the square-free d or
- the field discriminants.

It seems the field discriminants are either the square-free d,
or 4d.

For d = -19, the class number is 1. The Weber function

f(z) is given as: exp(-pi i/24) eta((1+z)/2)/eta(z).

Muzaffar and Williams have their main result in Theorem 3
and an example in Theorem 4, that
f(sqrt(-19)) = theta, where theta is the unique real
root of x^3 -2x - 2 = 0.

I have experimented with z = -29 (class number 6, discriminant= -116)
and z = -647.

So far, it seems that (i) f(z) is an algebraic integer, and that
(ii) the degree of f(z) over Q is a multiple of the class number.

By using an integer-relation algorithm implemented as
lindep() in PARI-gp, f(I*sqrt(29)) seems to be an algebraic integer
of degree 24 = 4*6, and f(I*sqrt(647)) seems to be an algebraic
integer of degree 46 = 2*23 with (maybe) the minimal polynomial:

a^46 -788*a^44 +6324*a^42 -11800*a^40 -243552*a^38 -802880*a^36 -1138880*a^34
-2161408*a^32 -5906432*a^30 -11246080*a^28 -8573952*a^26 -13555712*a^24
-25296896*a^22 -29081600*a^20 -11206656*a^18 -14221312*a^16 -22216704*a^14
-58195968*a^12 +6029312*a^10 + 42467328*a^8 -30408704*a^6 -48234496*a^4 +
41943040*a^2 - 8388608 .


Another thing I don't understand is what do modular equations
such as f(I*sqrt(19)) = theta (as above) tell us?

David Bernier

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