Joint results 2002

of Karim Belabas, Aïssa Derhem, and Daniel C. Mayer

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On these pages, we present most recent results of our joint research, directly from the lab.
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Basic bibliography:
K. Belabas, A fast algorithm to compute cubic fields, Math. Comp. 66 (1997), 1213-1237
A. Derhem, Capitulation dans les extensions quadratiques de corps de nombres cubiques cycliques, Thèse de doctorat, Université Laval, Quebec, 1988
D. C. Mayer, Multiplicities of dihedral discriminants, Math. Comp. 58 (1992), 831-847 and S55-S58
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E-mail addresses:
Karim.Belabas@math.u-psud.fr
aderhem@yahoo.fr
danielmayer@algebra.at
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Minimal occurrences of cubic multiplicities (2002/02/05)
Dan (02/01/31): I am very happy that you confirmed my old examples for multiplicity 5,
which was unknown up to now (except to Pierre Barrucand).
I found the pure cubic example -16008300 in Winnipeg City (1990)
and the totally real ones (starting with +89407866420) in Bregenz (1992).
I conjecture they are minimal.
Karim (02/01/31): Just checked with CCF up to 10^9:

-23 is minimal for m = 1
-972 ...m = 2
-1228 ...m = 3
-3299 ...m = 4
-16008300 ...m = 5 [conjecture is correct]
-70956 ...m = 6

-1190700 ...m = 8
-274347 ...m = 9

-5856300 ...m = 12
-3321607 ...m = 13

-144074700 ...m = 16

-27434700 ...m = 18
-167644728 ...m = 27

-653329427 ...m = 40
Karim: And with CRF up to 4.10^10:

49 is minimal for m = 1
3969 ...m = 2
22356 ...m = 3
32009 ...m = 4
13302897300 ...m = 5 [conjecture was incorrect]
3054132 ...m = 6

242144721 ...m = 8
18251060 ...m = 9

998997300 ...m = 12
39345017 ...m = 13

942216948 ...m = 18

3700879348 ...m = 27
("holes" do not occur as a multiplicity so far)
Dan (02/02/02): Fantastic, Karim !
Your new CCF/CRF seems to be incredibly powerful.
With these results you have turned my dreams into truth !
It is really a hilite in our coop.
Let me add some history, comments, and analysis:

A. Complex cubic fields L, |d_L|<10^9


-------------------------------------------------------------------------------
m | d_L | computed by | analyzed by | f | d_K | congruent | r | s | t | w | v | o |
-------------------------------------------------------------------------------
1 | -23 | Angell, 1972 | | 1 | -23 | | 1 | 1 | 0 | 0 | 0 | 0c |
2 | -972 | Angell, 1972 | | 3^2*2 | -3 | -3(mod 9) | 0 | 1 | 1 | 2 | 2 | 1i |
3 | -1228 | Angell, 1972 | | 2 | -307 | | 1 | 1 | 1 | 0 | 0 | 0f |
4 | -3299 | Angell, 1972 | | 1 | -3299 | | 2 | 2 | 0 | 0 | 0 | 0c |
5 | -16008300 | Mayer, 1989 | Mayer, 1990 | 2*3*5*7*11 | -3 | -3(mod 9) | 0 | 1 | 4 | 1 | 5 | 1r |
6 | -70956 | Fung/Williams, 1990 | Mayer, 1990 | 3^2*2 | -219 | -3(mod 9) | 0 | 0 | 1 | 2 | 0 | 0i |
7 | --- | | | | | | | | | | | |
8 | -1190700 | Mayer, 1989 | Mayer, 1990 | 3^2*2*5*7 | -3 | -3(mod 9) | 0 | 1 | 3 | 2 | 4 | 1i |
9 | -274347 | Fung/Williams, 1990 | Mayer, 1990 | 3^2 | -3387 | -3(mod 9) | 1 | 1 | 0 | 2 | 0 | 0i |
10 | ??? | | | | | | | | | | | |
11 | ??? | | | | | | | | | | | |
12 | -5856300 | Belabas, 1997 | | 3^2*2*5 | -723 | -3(mod 9) | 0 | 0 | 2 | 2 | 0 | 0i |
13 | -3321607 | Diaz y Diaz, 1973 | Buell, 1975 | 1 | -3321607 | | 3 | 3 | 0 | 0 | 0 | 0c |
| | | &Shanks, 1976 | | | | | | | | | |
14 | --- | | | | | | | | | | | |
15 | ??? | | | | | | | | | | | |
16 | -144074700 | Mayer, 1989 | Mayer, 1990 | 3^2*2*5*7*11 | -3 | -3(mod 9) | 0 | 1 | 4 | 2 | 5 | 1i |
17 | --- | | | | | | | | | | | |
18 | -27434700 | Belabas, 1997 | | 3^2*2*5 | -3387 | -3(mod 9) | 1 | 1 | 2 | 2 | 2 | 1i |
* | *** | | | | | | | | | | | |
27 | -167644728 | Belabas, 1997 | | 3^2 | -2069688 | -3(mod 9) | 2 | 2 | 0 | 2 | 0 | 0i |
* | *** | | | | | | | | | | | |
40 | -653329427 | Diaz y Diaz, 1978 | Belabas, 2002 | 1 | -653329427 | | 4 | 4 | 0 | 0 | 0 | 0c |
-------------------------------------------------------------------------------

Notation:

r denotes the 3-class rank of K,
s the modified 3-class rank (as in the Vienna Congress presentation, r <= s <= r+1),
t = #{ q prime | q != 3, v_q(f) >= 1 },
w = 0, if v_3(f) = 0,
w = 2, if v_3(f) = 2, d_K congruent -3(mod 9),
and w = 1, otherwise, as in my previous emails,
v the number of all restrictive ("bad") prime divisors of f (including 3 or 3^2, if suitable), and
o the order of the relevant multiplicity formula, with a postfix indicating the type of contribution
(c...class rank, r...regular conductor, i...irregular conductor)

A'. Supplement with 10^9 < |d_L| < 10^{10}

-------------------------------------------------------------------------------
m | d_L | computed by | analyzed by | f | d_K | congruent | r | s | t | w | v | o |
-------------------------------------------------------------------------------
10 |-4626398700 | Mayer, 1990 | | 2*3*5*7*11*17| -3 | -3(mod 9) | 0 | 1 | 5 | 1 | 5 | 1r |
11 |-2705402700 | Mayer, 1990 | | 2*3*5*7*11*13| -3 | -3(mod 9) | 0 | 1 | 5 | 1 | 6 | 1r |
-------------------------------------------------------------------------------

It is illuminating (historically and theoretically) to arrange the cases by the order of the multiplicity formula:

A0c. Unramified (Hilbert) class fields with fundamental multiplicities m_3(d_K,1) = (3^r - 1) / 2 (Hasse, 1929)

-------------------------------------------------------------------------------
m | d_L| computed by | analyzed by | f | d_K| congruent | r | s | t | w | v | o |
-------------------------------------------------------------------------------
1 | -23| Angell, 1972 | | 1 | -23| | 1 | 1 | 0 | 0 | 0 | 0c |
4 | -3299| Angell, 1972 | | 1 | -3299| | 2 | 2 | 0 | 0 | 0 | 0c |
13 | -3321607| Diaz y Diaz, 1973 | Buell, 1975 | 1 | -3321607| | 3 | 3 | 0 | 0 | 0 | 0c |
40 | -653329427| Diaz y Diaz, 1978 | Belabas, 2002 | 1 | -653329427| | 4 | 4 | 0 | 0 | 0 | 0c |

121 |-5393946914743| Quer? | | 1 |-5393946914743| | 5 | 5 | 0 | 0 | 0 | 0c |
-------------------------------------------------------------------------------

A0r. Hasse's free regular multiplicity formula of order 0: m_3(d_K,f) = 3^r*2^{t+w-1}
A0i. Hasse's free irregular multiplicity formula of order 0: m_3(d_K,f) = 3^{r+1}*2^t (both 1929)

-------------------------------------------------------------------------------
m | d_L | computed by | analyzed by | f | d_K | congruent | r | s | t | w | v | o |
-------------------------------------------------------------------------------
3 | -1228 | Angell, 1972 | | 2 | -307 | | 1 | 1 | 1 | 0 | 0 | 0r |
6 | -70956 | Fung/Williams, 1990 | Mayer, 1990 | 3^2*2 | -219 | -3(mod 9) | 0 | 0 | 1 | 2 | 0 | 0i |
9 | -274347 | Fung/Williams, 1990 | Mayer, 1990 | 3^2 | -3387 | -3(mod 9) | 1 | 1 | 0 | 2 | 0 | 0i |
12 | -5856300 | Belabas, 1997 | | 3^2*2*5 | -723 | -3(mod 9) | 0 | 0 | 2 | 2 | 0 | 0i |
27 | -167644728 | Belabas, 1997 | | 3^2 | -2069688 | -3(mod 9) | 2 | 2 | 0 | 2 | 0 | 0i |
-------------------------------------------------------------------------------

A1r. Mayer's restrictive regular multiplicity formula of order 1:
m_3(d_K,f) = 3^r*2^u*(2^{v-1}-(-1)^{v-1})/3 with u=t+w-v
A1i. Mayer's restrictive irregular multiplicity formulas of order 1:
m_3(d_K,f) = 3^r*2^t, if delta_3(3) = 1,
and m_3(d_K,f) = 3^{r+1}*2^u*(2^{v-1}-(-1)^{v-1})/3 with u = t+1-v, if delta_3(3) = 0 (all 1990)

-------------------------------------------------------------------------------
m | d_L | computed by | analyzed by | f | d_K | congruent | r | s | t | w | v | o |
-------------------------------------------------------------------------------
2 | -972 | Angell, 1972 | | 3^2*2 | -3 | -3(mod 9) | 0 | 1 | 1 | 2 | 2 | 1i |
5 | -16008300 | Mayer, 1989 | Mayer, 1990 | 2*3*5*7*11 | -3 | -3(mod 9) | 0 | 1 | 4 | 1 | 5 | 1r |
8 | -1190700 | Mayer, 1989 | Mayer, 1990 | 3^2*2*5*7 | -3 | -3(mod 9) | 0 | 1 | 3 | 2 | 4 | 1i |
10 |-4626398700 | Mayer, 1990 | | 2*3*5*7*11*17| -3 | -3(mod 9) | 0 | 1 | 5 | 1 | 5 | 1r |
11 |-2705402700 | Mayer, 1990 | | 2*3*5*7*11*13| -3 | -3(mod 9) | 0 | 1 | 5 | 1 | 6 | 1r |
16 | -144074700 | Mayer, 1989 | Mayer, 1990 | 3^2*2*5*7*11 | -3 | -3(mod 9) | 0 | 1 | 4 | 2 | 5 | 1i |
18 | -27434700 | Belabas, 1997 | | 3^2*2*5 | -3387 | -3(mod 9) | 1 | 1 | 2 | 2 | 2 | 1i |
-------------------------------------------------------------------------------

B. Totally real cubic fields L, d_L < 4*10^{10}

-------------------------------------------------------------------------------
m | d_L | computed by | analyzed by | f | d_K | congruent | r | s | t | w | v | o |
-------------------------------------------------------------------------------
1 | 49 | Godwin/Samet, 1959 | | 7 | (K=Q) 1 | | 0 | 0 | 1 | 0 | 0 | 0r |
2 | 3969 | Godwin/Samet, 1959 | | 3^2*7 | (K=Q) 1 | | 0 | 0 | 1 | 1 | 0 | 0r |
3 | 22356 | Angell, 1975 | | 3^2*2 | 69 | -3(mod 9) | 0 | 1 | 1 | 2 | 2 | 1i |
4 | 32009 | Angell, 1975 | | 1 | 32009 | | 2 | 3 | 0 | 0 | 0 | 0c |
5 |13302897300 | Belabas, 2002 | | 3^2*2*5*7*11 | 277 | 1(mod 3) | 0 | 1 | 4 | 1 | 5 | 1r |
6 | 3054132 | Llorente/Quer, 1988 | Mayer, 1990 | 2*3 | 84837 | 3(mod 9) | 1 | 2 | 1 | 1 | 0 | 0r |
7 | --- | | | | | | | | | | | |
8 | 242144721 | Belabas, 1997 | | 3^2*7*13*19 | (K=Q) 1 | | 0 | 0 | 3 | 1 | 0 | 0r |
9 | 18251060 | Belabas, 1997 | | 2 | 4562765 | | 2 | 3 | 1 | 0 | 0 | 0r |
10 | ??? | | | | | | | | | | | |
11 | ??? | | | | | | | | | | | |
12 | 998997300 | Belabas, 1997 | | 3^2*2*5*7 | 2517 | -3(mod 9) | 0 | 1 | 3 | 2 | 2 | 1i |
13 | 39345017 | ?probably earlier? | Belabas, 2002 | 1 | 39345017 | | 3 | 4 | 0 | 0 | 0 | 0c |
* | *** | | | | | | | | | | | |
18 | 942216948 | Belabas, 1997 | | 3^2*2 | 2908077 | -3(mod 9) | 1 | 2 | 1 | 2 | 0 | 0i |
* | *** | | | | | | | | | | | |
27 | 3700879348 | Belabas, 1997 | | 2 | 925219837 | | 3 | 4 | 1 | 0 | 0 | 0r |
-------------------------------------------------------------------------------

B0c. Unramified (Hilbert) class fields with fundamental multiplicities m_3(d_K,1) = (3^r - 1) / 2 (Hasse, 1929)

-------------------------------------------------------------------------------
m | d_L | computed by | analyzed by | f | d_K | congruent | r | s | t | w | v | o |
-------------------------------------------------------------------------------
4 | 32009 | Angell, 1975 | | 1 | 32009 | | 2 | 3 | 0 | 0 | 0 | 0c |
13 | 39345017 | ?probably earlier? | Belabas, 2002 | 1 | 39345017 | | 3 | 4 | 0 | 0 | 0 | 0c |
-------------------------------------------------------------------------------

B0f. Hasse's free cyclic multiplicity formula of order 0: m_3(f) = 2^{t+w-1}

-------------------------------------------------------------------------------
m | d_L | computed by | analyzed by | f | d_K | congruent | r | s | t | w | v | o |
-------------------------------------------------------------------------------
1 | 49 | Godwin/Samet, 1959 | | 7 | (K=Q) 1 | | 0 | 0 | 1 | 0 | 0 | 0f |
2 | 3969 | Godwin/Samet, 1959 | | 3^2*7 | (K=Q) 1 | | 0 | 0 | 1 | 1 | 0 | 0f |
8 | 242144721 | Belabas, 1997 | | 3^2*7*13*19 | (K=Q) 1 | | 0 | 0 | 3 | 1 | 0 | 0f |
-------------------------------------------------------------------------------

B0r. Hasse's free regular multiplicity formula of order 0: m_3(d_K,f) = 3^r*2^{t+w-1}
B0i. Hasse's free irregular multiplicity formula of order 0: m_3(d_K,f) = 3^{r+1}*2^t (both 1929)

-------------------------------------------------------------------------------
m | d_L | computed by | analyzed by | f | d_K | congruent | r | s | t | w | v | o |
-------------------------------------------------------------------------------
6 | 3054132 | Llorente/Quer, 1988 | Mayer, 1990 | 2*3 | 84837 | 3(mod 9) | 1 | 2 | 1 | 1 | 0 | 0r |
9 | 18251060 | Belabas, 1997 | | 2 | 4562765 | | 2 | 3 | 1 | 0 | 0 | 0r |
18 | 942216948 | Belabas, 1997 | | 3^2*2 | 2908077 | -3(mod 9) | 1 | 2 | 1 | 2 | 0 | 0i |
27 | 3700879348 | Belabas, 1997 | | 2 | 925219837 | | 3 | 4 | 1 | 0 | 0 | 0r |
-------------------------------------------------------------------------------

B1r. Mayer's restrictive regular multiplicity formula of order 1:
m_3(d_K,f) = 3^r*2^u*(2^{v-1}-(-1)^{v-1})/3 with u = t+w-v
B1i. Mayer's restrictive irregular multiplicity formulas of order 1:
m_3(d_K,f) = 3^r*2^t, if delta_3(3) = 1,
and m_3(d_K,f) = 3^{r+1}*2^u*(2^{v-1}-(-1)^{v-1})/3 with u=t+1-v, if delta_3(3) = 0 (all 1990)

-------------------------------------------------------------------------------
m | d_L | computed by | analyzed by | f | d_K | congruent | r | s | t | w | v | o |
-------------------------------------------------------------------------------
3 | 22356 | Angell, 1975 | | 3^2*2 | 69 | -3(mod 9) | 0 | 1 | 1 | 2 | 2 | 1i |
5 |13302897300 | Belabas, 2002 | | 3^2*2*5*7*11 | 277 | 1(mod 3) | 0 | 1 | 4 | 1 | 5 | 1r |
12 | 998997300 | Belabas, 1997 | | 3^2*2*5*7 | 2517 | -3(mod 9) | 0 | 1 | 3 | 2 | 2 | 1i |
-------------------------------------------------------------------------------

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