Attach the right witness to the 857.567 record in 87a

[Chessing234]

Closes#150 (reported by @u00dxk2, who did the diagnosis).

The [HMR2019] row of constants/87a.md quoted

$\mathrm{rd} \le 3^4\cdot 5^4\cdot 7^4\cdot 13^2\cdot 29^4\cdot 53^2\cdot 109^2 \le 857.5662\dots$

That product is 484887097019201067725625, a 24-digit integer, so it cannot be a root discriminant. It is also not the witness behind this record.

I checked both halves against the paper itself (ar5iv 1901.04354) rather than taking the issue's word for it:

• §3.3.1, "An example of J. Martin" — a degree-8 field whose discriminant is exactly that factorization, "and the root discriminant is less than 913.4927". 484887097019201067725625^(1/8) = 913.492694…, which is the 913.493 on the [Mar2006] row directly above. So the factorization belongs one row up.
• §3.3.3, "The totally real case" — a different, degree-12 field with rd_K < 770.6432, cut at a single prime above 13 of norm 13, giving rd_{K_S^[1]} = rd_K · 13^(1/12·(1−1/2)) < 857.5662, which the paper calls "a new record with savings by a factor of 13^{1/24}". 770.6432 · 13^(1/24) = 857.566203….

So the record value 857.567 is right, but the explicit witness beside it — the factorization and the "8-th root … rank 8" description — came from §3.3.1. §3.3.3's field has d_2 G = 9, not rank 8.

The row now carries the degree-12 construction that actually produces the bound, and the factorization moves up to the [Mar2006] row where its 8th root is the quoted value. No bound changes.

Note @u00dxk2 offered to open this themselves and asked which wording you'd prefer — happy for this to be closed in favour of theirs if you'd rather, or reworded here.

This was AI-assisted; the two arithmetic checks and both quotations were verified against the arXiv full text.

[teorth]

Merged, thanks — and thanks to @u00dxk2 for the diagnosis in #150.

I checked this against the primary source rather than the issue, and every claim holds. From [HMR2019] §3.3.1, on Martin's degree-8 field: "Its discriminant is $3^4\cdot 5^4\cdot 7^4\cdot 13^2\cdot 29^4\cdot 53^2\cdot 109^2$ and the root discriminant is less than 913.4927" — and I get $484887097019201067725625^{1/8} = 913.4926943\ldots$, so the factorization does belong on the [Mar2006] row. From §3.3.3, the totally real case: "Let K be the totally real field of [14] of degree 12 over $\mathbb{Q}$ ... and $rd_K < 770.6432$", then $\mathrm{rd}_{K_S^{[1]}} = \mathrm{rd}K\cdot 13^{\frac{1}{12}(1-\frac12)} < 857.5662$, "a new record with savings by a factor of $13^{1/24}$" — and $770.6432\cdot 13^{1/24} = 857.5662030\ldots$. The paper also records $d_2 G\emptyset = d_2 G_S = 9$ for that field, which confirms the "rank 8" in the old row came from the degree-8 example.

So the row was conflating §3.3.1 and §3.3.3, exactly as diagnosed. This was my error when I wrote the page — the value was right and the witness beside it was not, which is the more embarrassing way round.

One thing you might add as a follow-up if you feel like it: §3.3.4 notes that if $\tau_{\mathfrak{p}_{13}}$ had depth greater than two in $G_S$, the same field would give an infinite $2$-Hilbert class field tower and the totally real record would drop to $&lt; 770.644$ — worth a line in "Additional comments and links" as the nearest visible improvement, though it is a hope rather than a bound.

[Chessing234]

Thanks. Follow-up opened as #152 — a bullet in "Additional comments and links" for §3.3.4, with the §5.1 pointer that the paper sends you to, and no table or README change since it is a hope rather than a bound.