Other Heegner Numbers
For the four largest Heegner numbers, the approximations one obtains are as follows.
Alternatively,
where the reason for the squares is due to certain Eisenstein series. For Heegner numbers, one does not obtain an almost integer; even is not noteworthy. The integer j-invariants are highly factorisable, which follows from the form, and factor as,
These transcendental numbers, in addition to being closely approximated by integers, (which are simply algebraic numbers of degree 1), can also be closely approximated by algebraic numbers of degree 3,
The roots of the cubics can be exactly given by quotients of the Dedekind eta function η(τ), a modular function involving a 24th root, and which explains the 24 in the approximation. In addition, they can also be closely approximated by algebraic numbers of degree 4,
Note the reappearance of the integers as well as the fact that,
which, with the appropriate fractional power, are precisely the j-invariants. As well as for algebraic numbers of degree 6,
where the xs are given respectively by the appropriate root of the sextic equations,
with the j-invariants appearing again. These sextics are not only algebraic, they are also solvable in radicals as they factor into two cubics over the extension (with the first factoring further into two quadratics). These algebraic approximations can be exactly expressed in terms of Dedekind eta quotients. As an example, let, then,
where the eta quotients are the algebraic numbers given above.
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