# Fung's Table of Complex Cubic Fields

## The first families with multiplicity greater than 4

A voluminous PhD thesis

In 1989 G. W. Fung computed a large table of complex cubic fields L with discriminants dL > -106 as part of a PhD thesis [0] written under the supervision of H. C. Williams. He sent me a preprint of statistical results concerning this table on the 19th of December 1989. One of the most interesting features was the occurrence of 7 families of fields with multiplicity 5. I asked Fung immediately to communicate the discriminants and generating polynomials of these fields. But, unfortunately, I did not get any response. The lack of numerical information caused me to start detailed investigations into multiplicities of cubic discriminants.

Do there exist 5-families of non-pure complex cubic fields?

During the first half of 1990 I developed an essentially improved version of my own algorithm for computing complex cubic fields. The original version had been the tool for constructing my table [1] of complex cubic fields L with discriminants dL > -30000 and their principal factorization types in August 1989. I used the improved algorithm to extend this table and to extract some 6-families in July 1990. Meanwhile, Fung's report [2] had appeared in "Mathematics of Computation" in July 1990.

On the 30th of August 1990 I had the opportunity to have a glance at Fung's table in Winnipeg City. I had browsed through the 535 sheets of very dense double sided computer output for only about 5 minutes and, wow, there it was: the first 5-family with the smallest discriminant. It consisted of 5 cyclic cubic extensions with conductor f = 18 over the quadratic field with discriminant dK = -291. These S3-fields contain 5 complex cubic fields sharing the same discriminant dL = -94284, a value that sounded rather familiar to me. Indeed, a comparison with my well prepared 6-families revealed the lack of the field with the largest regulator 25.6 in this 6-family in Fung's table. The following diagram summarizes some properties of the members of this 6-family and emphasizes the missing field with red color. The fields are ordered by their regulators Reg. We give the coefficients C and D of a generating polynomial P(X) = X3 - CX - D, the index I(P) of P(X), the index of the unit group UO of the suborder O generated by the real zero of P(X) in the unit group U of the maximal order, and one of the principal factors PF (the other is the square of PF).

Reg C D I(P) (U:UO) PF
8.9 -36 84 2 2 2
11.3 54 234 3 1 3
13.1 18 66 1 1 6
13.5 36 102 1 1 2
23.8 -18 294 5 2 6
25.6 -54 90 3 1 3

Recomputation due to my invaluable hints

Without my generating polynomial P(X) = X3 + 54X - 90 Fung would never have detected the error in an assembly language subroutine of his extensive computer program. The fact was that there were no 5-families at all in the whole table. The publication of the table errata [3] was intentionally delayed until the minds were calmed already in 1994. It would have been adequate to mention my essential cooperation concerning the elimination of the 669 errors, but it was intentionally neglected. In the opinion of H. C. Williams the theoretical aspects of [2] were not affected. However, my impression is that the careless handling of mysterious discriminantal multiplicities was a serious offence against theoretical aspects, indeed.

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