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I. Condition

If there are integers $(r_1, r_2, r_3, r_4)$ such that,

$$ar_1^2+br_1r_2+cr_2^2=ac\\ r_3=(ar_1+br_2)/c\\ r_4=(br_1+cr_2)/a$$

then $(a u^2+buv+cv^2)(a x^2+bxy+cz^2)= (a z_1^2+bz_1z_2+cz_2^2)$, or the quadratic form $(a,b,c)$ has multiplicative closure over the integers. (Proof is in this MO answer.)

In that previous post, we gave examples of negative discriminant $d$ with class number $h(d) = 3m$ where at least three reduced quadratic forms have closure namely, $(1,b_1,c_1), (a,b,c), (a,-b,c)$. However, its seems the interesting subset is when $h(d) = 9m$ where it can now be more than three forms. In the lists below for imaginary quadratic fields, assume the class group of $\mathbb Q(\sqrt{-d})$, and $d$ is negated for convenience.

A. Class number 9 (A046006)

Isomorphic to $C_3 \times C_3$ only for $d = 4027$. The rest is $C_9$.

Note: So $h(-4027) = 9$ is the unique $d$ such that all $9$ reduced quadratic forms,

$$(1, 1, 1007), (13, -9, 79), (13, 9, 79), (17, -11, 61), (17, 11, 61), (19, -1, 53), (19, 1, 53), (29, -27, 41), (29, 27, 41)$$

have multiplicative closure.

Note: Does this $d$ have any other unique properties? A possibility is $a(9)=4027$ of "Number of cusps in a class of degree-$3n$ complex algebraic surfaces". (A225018)

B. Class number 18 (A046015)

Isomorphic to $C_3 \times C_6$ for $d = 9748, 12067, 16627, 17131, 19651, 22443, 23683, 34027, 34507$. The rest is $C_{18}$.

Note: These $d$ have $18$ reduced quadratic forms with $9$ having multiplicative closure.

C. Class number 27 (A351665)

Isomorphic to $C_3 \times C_9$ for $d = 3299, 19427, 34603, 89923, 98443$. The rest is $C_{27}$.

Note: Similar for these $d$.

D. Class number 36 (A351674)

Isomorphic to $C_3 \times C_{12}$ (for 16 $d$) or $C_6 \times C_6$ (for 23 $d$) so $16+23=39$ $d$ (odd and even). Unfortunately, OEIS does not identify these.

II. Question

So for the last sequence, I tested all 668 entries (well, the odd $d$). Only the following $26$ odd $d$ have $9$ quadratic forms with multiplicative closure,

$$6583, 12131, 18555, 19187, 22395, 26139, 27355, 34867, 35539, 37219, 42619, 43827, 45835, 46587, 48667, 54195, 81867, 83395, 92827, 93067, 95155, 112795, 114403, 116083, 133555, 205363$$

Q: Which of these are isomorphic to $C_3 \times C_{12}$ and which to $C_6 \times C_6$?

P.S. For $27$ quadratic forms with multiplicative closure, I assume a necessary (but not sufficient) condition is $h(d)=27m$ (though $m=1$ is ruled out). The smallest so far is $h(-d) = 27\cdot6 =162$ for $d=3640387$. (See A244574, or $d$ with 3-class rank 3, all of which have $h(-d)=27m$.)

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    $\begingroup$ Not enough to be an answer so I'm posting this as a comment. If you have PARI/GP you can simply run for(d=1, oo, if(isfundamental(-d) && quadclassunit(-d)[2] == [12,3], print1(d, ", "))) and wait untill all $16$ numbers occur (since you know there are only $16$). I got $d=3896, 6583, 12131, 19187, 23428, 26139, 27355, 35539, 39736, 43192, 48667, 93067, 95155, 114403, 133555, 205363$. (Ah, I saw that it was said that the largest number in A351674 is 217627, so you could replace the "oo" in the code by this number). $\endgroup$ Commented Jan 7 at 20:53
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    $\begingroup$ @JianingSong I think they would make two nice OEIS entries, won’t they? :) Considering they are linked to multiplicative closure. $\endgroup$ Commented Jan 7 at 23:07
  • $\begingroup$ Yes I do agree, please go ahead :) $\endgroup$ Commented Jan 12 at 11:32
  • $\begingroup$ @JianingSong Well, if Daniel C. Mayer made about two dozen sequences all involving the 3-class group, then these two should be a nice addition. By the way, if I may ask, would be the Pari command if I wish a list of all 228 fundamental $-d$ with class number 81? $\endgroup$ Commented Jan 12 at 13:22
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    $\begingroup$ Certainly, you replace part of the code by quadclassunit(-d)[1] == 81 :) $\endgroup$ Commented Jan 13 at 19:32

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Courtesy of Jianing Song's code,

for(d=1, 217627, if(isfundamental(-d) && quadclassunit(-d)[2] == [12,3], print1(d, ", ")))

in the comments and the online PARI/GP calculator, then for $h(-d)=36$,

A. 16 $d$ with class group isomorphic to $C_{12} \times C_{3}$

$(3896,6583,12131,19187,23428,26139,27355,35539,39736,43192,48667,93067,95155,114403,133555,205363)$

B. 23 $d$ with class group isomorphic to $C_6 \times C_6$

$(15544, 18555, 20568, 22395, 24340, 27640, 31908, 32968, 34867, 37219, 40692, 42619, 43827, 45835, 46587, 48052, 48472, 54195, 81867, 83395, 92827, 112795, 116083)$

So $16+23 = 26\,\text{(odd)}+13\,\text{(even)}=39$, and it is satisfying my method didn't miss any odd $d$. (For some reason, my old Mathematica doesn't give the classlist for even $d$.)

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    $\begingroup$ Using Jianing Song's code for the next square $h(-d)=9m=81$, then \begin{align} d &= 45131\\ d &= 134059\end{align} have class group isomorphic to $C_{27}\times C_3$, and $C_9\times C_9$, respectively. Unfortunately, both only have $9$ quadratic forms with multiplicative closure. (I was hoping for $27$). So the smallest known still is $h(-d)=27m=162$ given in the post. $\endgroup$ Commented Jan 8 at 3:17

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