MATHEMATICS OF COMPUTATION
VOLUME 48, NUMBER 177
JANUARY 1987, PAGES 39-54
The Computation of the Fundamental Unit
of Totally Complex Quartic Orders
By Johannes Buchmann*
Dedicated to Professor Daniel Shanks on the occasion of his 10th birthday
Abstract. We describe an efficient algorithm for the computation of the regulator and a
fundamental unit of an arbitrary totally complex quartic order. We analyze its complexity and
4,-
we present tables with computational results for the orders Z[y - d ], 1 < d < 500.
1. Introduction. The computation of fundamental units in orders of algebraic
number fields is one of the main problems in computational algebraic number
theory.
The simplest fields for this problem are those with only one fundamental unit.
There are three types of such fields; the real quadratic, the complex cubic, and the
totally complex quartic fields.
It is well known that the fundamental unit of a real quadratic field can be
computed by means of the ordinary continued fraction algorithm, cf. [2, II, Section
7]. There are interesting refinements of this algorithm due to Shanks [16] and
Lenstra [13].
The fundamental unit of complex cubic fields can be computed using Voronoi's
generalized continued fraction algorithm, cf. [5], [18]. This algorithm was discussed
and improved in several interesting papers of Williams et al., cf. [20].
For totally complex quartic fields there are only a few results. If the field contains
a real quadratic subfield, the computation of a fundamental unit can be reduced to
the computation of the fundamental unit of the subfield, cf. [8].
For complex quartic fields containing an imaginary quadratic subfield, Scharlau
proved that a fundamental unit is a minimal solution of a certain relative Pell
equation, cf. [15], but he did not give a method for solving it.
For complex quartic fields containing quadratic subfields of class number one,
there are results due to Amara [1] and Lakein [9], [10].
More generally, the author proved that the generalized Voronoi algorithm (GVA),
developed in [3], yields a fundamental unit for any order of a totally complex quartic
field. The algorithm of Jeans [7] seems to have some similarities with this method.
Received June 13, 1985; revised June 11, 1986.
1980 MathematicsSubjectClassification.Primary 12A30,12C20.
*Supported by a Feodor Lynen research fellowship of the Alexander von Humboldt-Foundation.
Permanent address: Mathematisches Institut, Universität Düsseldorf, 4000 Düsseldorf, West Germany.
©1987 American Mathematical Society
0025-5718/87 $1.00 + $.25 per page
39
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40
JOHANNES BUCHMANN
In this paper we describe how to apply the GVA practically. We analyze its
complexity and prove that it yields a fundamental unit of any totally complex
quartic order in 0(RDe) binary operations (for every e > 0), where D is the
absolute value of the discriminant and R is the regulator of the order.
We establish an analogue of Galois' theorem on the symmetry of the continued
fraction expansion of the square root of a rational number. We conclude the paper
4,-
by presenting computational results for the orders Z[y - d ], 1 < d < 500.
The author is indebted to the referee for many helpful corrections and suggestions.
2. Notations. In this paper
L = Q(p) is a totally complex quartic algebraic number field,
o is a Q-isomorphism of L into C, different from the complex conjugation. For
£ e L we write
£<!>
= £, {W-a(É),
£(3>= f and €<4>=aW-
0 is an order of L,
D is the absolute value of the discriminant of 0,
R is the regulator of 0,
ux,..., <o4is a Z-basis of 0,
<o*,..., toj is the corresponding dual basis,
W = max{\w^\\l < i, A:<4},
W* = max{|<o£<0||l < i, k^4}.
We assume that
(2.1)
W^D1'2.
Such a basis can be computed using a basis reduction algorithm, e.g., [11], in the
Minkowski lattice corresponding to 0. It follows immediately that
(2.2)
W* < 6D.
For a (fractional) ideal a of 0 we fix
d(a) = min{d<= N|Ja
ç 0),
N(a) = norm of a.
3. The Method. We recall the main definitions and results of [3]. There, we
introduced the map
L^R2,
¿-É=(m2,io(oi2)'
which, restricted to the multiplicative group Lx, is a homomorphism, if we use the
product
(yi,y2)' •{y^yí)'=
(^.a>íí)'-
Moreover, L x acts on L by
£*4' = 44' for every¿eLx,
4'g L.
For a point y = (yx, y2)' e R2 its norm is defined by
MjOHj'i^INow let a be a (fractional) ideal in 0. Then the image a is a discrete set in R2. A
point 0 #=m in a is called a minimal point of a, if its norm body
Q(m) = [ye
R2|0 < .y,< m,;for 1 < / < 2}
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FUNDAMENTAL UNIT OF TOTALLY COMPLEXQUARTIC ORDERS
41
does not contain points of a aside from 0 and m. Minimal points are of bounded
norm
(3.1)
JV(m)< (4/tr2)Dl/2N(a).
Moreover, for {u, v] = {1,2} the u-neighbor of a minimal point in is defined to be
the (uniquely determined) minimal point m' with m'u< mv and minimal m'u. For
this neighbor we have by Minkowski's convex body theorem
(3.2)
m'umv^(4/ir2)Dl/2N(a).
Finally, 1 is a minimal point in 0 and all the minimal points of 0 can be arranged in
a two-sided sequence (mk)keZ, m0 = 1, where mk+x is always the 2-neighbor of mk
and, in turn, mkis the 1-neighbor of mk+x for every k e Z. This sequence, called
the GVA-expansion in 0, is of the purely periodic form
...,e_1*l,
e~1*mx,...,e~um
(3.3)
,, 1, mx,...,m
,,
£*1, e*m1,...,e*m/,_1,....
Here e is a unit in 0, and if p is chosen minimal, then e is a fundamental unit of 0,
and p is called the period length of the GVA in 0.
4. The Algorithm. Here is our algorithm for computing a fundamental unit e and
the regulator R of 0.
Algorithm
4.1.
Input: «j,...,to4.
Output: R, e.
1. Initialize: k «- 0, N *- 1, t/0 <- 1, a <- 0, .R <- 0.
2. Repeat until A:^ 1 and TV= 1:
(a) a «- (l/vk)a(b) Compute r/^+1 in a such that T)k+1is the 2-neighbor of 1 in a.
(c)k+-k
+ l, R+-Rlog|T,,|2, N+-N- \NLlQ(vk)\-
3.e+-nf.0r,j.
Notice that we can compute a maximal system of pairwise nonassociated "minima"
in the sense of [21, Section 3], if we calculate in step 2(b) all the ijin a such that t| is
the 2-neighbor of 1 in a.
The representation of the principal ideals a will be discussed in Section 6 and the
computation of r\k+x will be described in Section 7.
We conclude this section by giving a justification for our algorithm. We define for
k
(4.1)
M*
= 7rk=0
Then we have
(4.2)
pk e 0
and
\ik = rhk for 0 < k < p,
and in step 2(b)
(4.3)
a = (l/pk)0.
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42
JOHANNES BUCHMANN
In fact, (4.2) and (4.3) are true for k = 0. Now suppose that (4.2) and (4.3) hold for
k > 0. Since -qk+xe (l/pk)0, we must have pk+x = pkr¡k+x e 0. Moreover, T)k+I is
the 2-neighbor of 1 in (l/uk)0, and therefore (ik+1 = ukT|k+i must be the 2-neighbor of (ik = mk in 0, i.e., jik+1 = wA+1. It follows immediately that in step 2(c),
(4-4)
N = N(mk)=\NLlQ(pk)\.
If k = p, then by (3.3) and (4.4) we must have N = 1, and because of the minimality
of p, this is the first time that 7Y= 1 can happen.
5. The Theorem of Galois. By a theorem of Galois, the period of the continued
fraction expansion of the square root of a positive rational number is symmetric, cf.
[14, Section 23]. A similar result is proved in this section. We assume that the order
under consideration satisfies the condition
(5.1)
a(0) = 0.
4,-
This is true, for example, if 0 = Z[\/-d], d e N. Note that (5.1) implies that L has
a quadratic subfield. On the plane R2 we introduce the reflection
(5.2)
d:R2-*R2,
y = (yi,y2y-*S(y)
= (y2,yiy.
Then we have for every £ e L
o(€)-*(€).
Consequently, the minimal points in 0 have the symmetry property:
Proposition
5.1. For every k e Z we have mk = a(m_k).
Proof. Let m, m' he minimal points of 0. Then a(m) and a(m') are minimal
points of 0, and m is the 1-neighbor of m' if and only if ö(m) is the 2-neighbor of
ö(m').
D
This yields the following application: If we have computed m0,...,mk,
then we
know m_k,...,mk.
Hence, we can compute the regulator and a fundamental unit of
0 by computing only half the period of the GVA-expansion in 0. This is done by
Algorithm
5.2.
Input: »!,...,
w4.
Output: R, e.
1. Initialize: k «- 0, R «- 0, ij0 «- 1, a «- 0.
2. Repeat:
(a) o <- (1/77Ja.
(b) Compute a complete system of nonassociated tj in a such that i\ is the
2-neighbor of 1 in o. Choose one of these tj's to be r\k+x.
(c) R +- R - log|r,,+1|2.
(d) If a(a) = (l/r/)ct holds for one of the tj's of (b), then e <- r¡T[k^x(i\j/a(r\j))
and return.
(e) R «- R + log|a(rJjt+1)|2.
(f) If a((l/r¡k+1)a) = (l/n)a
for one of the r/'s of (b), then e «(lA(i7fc+i))njc_i(T//°(7ï>))' and retum-
(g) k+-k + l.
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FUNDAMENTAL UNIT OF TOTALLY COMPLEX QUARTIC ORDERS
43
For the description of the representation of the ideals a and the computation in
2(b), 2(d) and 2(f) we refer to Sections 6 and 7. We conclude this section by giving a
justification of Algorithm 5.2.
For
k
M*= YlVj,
0< £</>,
7= 0
one can prove as in Section 4 that
(5.3)
pk&0
and
\ik = mk,
and that in step 2(b), (d) and (f)
(5.4)
and that
a = (l/pk)0,
(5.5)
wlt+1 = T)*^k
for every r/ computed in step 2(b).
It follows by (5.1) and Proposition 5.1 that
(5-6)
o(|ik) = m_¿,
and that in step 2(d)
(5.7)
o(a) = (l/o(pk))0.
Now suppose that in step 2(d), a(a) = (1/t/)o. Since in 2(d)
(5.8)
e = T)pk/o(pk),
it follows from (5.4) and (5.7) that e is a unit in 0. Moreover, by (5.5) and (5.6),
(5.9)
e*m_k = mk + x.
Because of the minimality of the period length p, this can happen only if p \ 2 k + 1.
But if p = 2k + 1, then (5.9) holds for every fundamental unit e with |e| < 1. Let e
be such a fundamental unit. It follows from (5.5), (5.6) and (5.9) that
(5.10)
£ • a(pk)
= a ■r\k+ xpk
with a e L, a = 1. We set tj = ccr)k+xand see by (5.4), (5.7) and (5.10) that r/ e a,
ri = T|k+1, anda(a)
= (l/7))a.
Analogously, one can show that in step 2(f) one has a((l/t]k+x)a) = (l/r/)a only
if e is a unit and p\2k + 2 and, in turn, that this happens in fact if p = 2k + 2.
Then e is again a fundamental unit of 0.
6. Basis Reduction and Ideal Representation. In this section we describe how we
represent the ideals a in Algorithm 4.1 and Algorithm 5.2.
First of all, we recall some properties of the basis reduction algorithm of Lenstra,
Lenstra and Lovász [11]. Let bx,..., b„ he a basis of a lattice T in Z" and let B > 2,
\bj\ < B for 1 <;' «s «.** This algorithm yields in 0(nA logB) binary operations a
basis ax,..., a„ of T which satisfies
(6.1)
fl|ûv|<2"(n-1>/4det(r),
7-1
where det(F) is the determinant of T.
**| I denotes the Euclidean norm.
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44
JOHANNES BUCHMANN
Now let a be an ideal in 0, and let ax,...,
(6.2)
aj=
a4 be a Z-basis of a,
I 4
\
¿Z akjo,k\/d(a),
W=i
1<;<4.
/
Denote by T the lattice spanned by the columns of the integral matrix A = (akJ).
We call A an LLL-matrix of a, if the columns of A form a basis of T which is
reduced in the sense of [11, p. 516]. The ideals in Algorithm 4.1 and Algorithm 5.2
are represented in terms of their common denominator and an LLL-matrix. It has
turned out in our computational experience that these representing integers are
always small compared to the discriminant of a. We are also able to give bounds on
the denominators and the elements of the LLL-matrices. These bounds are polynomials in D, and this means that the number of digits of these integers is 0(log D).
This statement will be useful in our complexity analysis in Section 8.
Proposition
6.1. If A is an LLL-matrix of a, then the column vectors ax,...,a4
of
A satisfy
(6.3)
cxld(a)N(a)1/4
<|o,|<
c2d(a)N(a)1/A
***
for 1 < / < 4.
Proof. Let ax,...,a4
be defined as in (6.2). Then we have for 1 < j < 4,
\aVf\aV\2 =\Nm(aj)\> N(a).
Hence, we have |«J'J| > N(a)1/4 for i — 1 or /' = 2, and the first inequality of (6.3)
follows from (2.1), (6.2), and Cauchy's inequality, whereas the second one follows
from the first one and (6.1), since in this case det(T) = N(a). D
Corollary
6.2. If A is an LLL-matrix of a and if a,,...,
a4 is the corresponding
Z-basis of a, definedin (6.2), then we havefor 1 < i, j < 4,
|ay|<c37V(a)1/4.
Proof. This corollary follows from (2.1), (6.2) and Proposition 6.1.
D
Corollary
6.3. // a is one of the ideals, used in Algorithm 4.1 or Algorithm 5.2, if
A is an LLL-matrix of a with the columns a,, ...,a4,
and if a,,...,a4
is the
corresponding Z-basis of a, defined in (6.2), then we have for 1 < i, j < 4,
| a •| < c4 and
«j01 < c5.
Proof. It follows from (3.1), (4.2), (4.3), (5.3), and (5.4) that tf(a)< 1 and
d(a) < D1/2. D
The last result shows that the ideals a in Algorithm 4.1 and Algorithm 5.2 can be
represented by integral matrices which are—independently of k—of the same
"small" size.
•The numbers ck, k e N, are of the form uD", u, v > 0.
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FUNDAMENTAL UNIT OF TOTALLY COMPLEX QUARTIC ORDERS
45
We finally remark that the comparison of the ideals in Algorithm 5.2 step 2(d) and
2(f) can be carried out by comparing the denominators and Hermite normal forms
of the representation matrices.
7. The Neighbor Computation. In step 2(b) of Algorithm 4.1 we want to know a
number tj in the ideal a such that t¡ is the 2-neighbor of 1 in a. Following the
explanation given in Section 3, this means that we have to find tj in a with
(7.1)
ha)|2<l
and
\^\\(4/tr2)Dl/2N(a)
with minimal |r/(2)|2.
Let A = (akj)i<kj<,4
De an LLL-matrix of a and let ax,...,aA be the corresponding Z-basis of û, defined in (6.2). For x = (xx,...,xA)' e Z4 and 1 < / < 2
we write
4
(7.2)
/
4
4
, I>,«f= \ L W Z akjxj\/d(a) if**0,
TJ(,)(x) = { y= i
\k = i
,_1
/
DV*N(a)l/2
ifx = 0.
Then we can compute tj using
Procedure 7.1.
1. Initialize: x2 «- 0, /«- 2.
2. Repeat:
Try to find xx e Z4 satisfying
(7.3)
Wl)(xx)\<l
and |„<2>(*i)|<\^(x2)\/f.
If the search is successful, then set x2 «- xx,
else
if / = 2, then set / <- 1,
else return tj = tj(1)(jc2). □
Notice that for all x e Z4 with
\^(x)\<l
and
|T,<2>(x)|<(2/7r)D1/47V(û)1/2,
by the well-known dual basis argument [2, p. 403],
4
(7.4)
E akJXj ^4W*d(a)Dl/4N(a)1/2,
1 < k < 4.
7= 1
The comparisons in (7.3) have to be carried out using rational approximations ù^
to u{k\ 1 < k, i < 4. We must therefore discuss the question of how this is to be
done. Let X > 0 have the property
(7.5)
max{|akn - w^111< /, k < 4} < \.
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46
JOHANNES BUCHMANN
For x =£ 0, let tj(í)(3c) be the approximation
of t)(,)(jc) obtained by substituting
u^
by &P in (7.2), and let tj(2)(0) be a rational approximation of £>1/4 such that
|tj(2)(0) - D1/4| < X. Finally, we set for x e Z4
(7.6)
E akjXj /d(a)
8x(x) = '
ifje*0,
7= 1
X
if x = 0,
and
(7.7)
ô(x) = 51(f)(2i)1/4
+ ôi(i)).
Then it follows for every x, subject to (7.4), that
(7.8)
8(x)^8
= 8x(2D1/4 + 8x),
with
81 = 16\W*D1/*N(a)1/2,
and we have for every x e Z4, subject to (7.4), and 1 < / < 2,
(7.9)
||Tj(')(x)|2-|T,<"(f)|2|<ô(f)<5.
Hence, (7.3) can be true only if
(7.10)
\'V^(xx) |2 + \r,m(xx) |2 < 1 + |t)<2>(*2) |2//2 + 3«.
The solutions of (7.10) can be computed using an algorithm of Fincke and Pohst
[6] which yields all the integral solutions x of an inequality Q(x) < K, where Q is a
positive definite «-dimensional rational quadratic form and K ^ 0 is a real constant.
By (7.9) we are able to decide whether a solution of (7.10) satisfies (7.3) as long as
\\tfl)(xx)'\-i\>8(xx)
(7-11}
,
,2
,
and
,2
,
llfPOc,)! -W2)(x-2)\/f2\>8(xx)+ 8(x2).
This, in turn, is true if
(7.12)
h(1)(*i)|*l
and \^(xx)\*\^(x2)\/f,
and X is small enough. Notice that (7.12) can be tested by using [21, Proposition
2.2].
Concluding these remarks, we can carry out the search for xx in Procedure 7.1 in
the following way.
We enumerate the solutions of (7.10) using the algorithm of Fincke and Pohst [6].
If we find a solution xx, we check if jcj satisfies (7.4). If not, we reject xx as a
possible solution of (7.3). Otherwise, we check whether xx is subject to
!*»(*)f
< l-aOO
and W2)(xx)\2<\^(x2)\2/f2-8(xx)-8(x2),
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FUNDAMENTAL UNIT OF TOTALLY COMPLEX QUARTIC ORDERS
47
or
\vm(xx)\2
> 1 + 8(xx)
or
|Tj<2)(f1)|2>|Tj<2)(f2)|2//2
+ ô(f1)
+ «(^).
In the first case, we have found a solution of (7.3). In the second case, we must
reject xx as a possible solution of (7.3).
If neither the first nor the second case holds, a situation which we have never
encountered in any of our computations, then we check whether (7.12) is true. If the
answer is negative, then we must reject xx; otherwise, we have to increase the
precision of our approximation to w^, i.e., we have to decrease X. During our
computations we found the value X = 10 ~12 always to be sufficient.
If we have enumerated all the solutions of (7.10) without finding a solution of
(7.3), then no such solution exists.
In Algorithm 5.2 we need all the tj g a such that r\ is the 2-neighbor of 1 in a.
These numbers can be computed by a further application of the algorithm in [6].
For our complexity analysis in the next section, it is necessary to be able to prove
that there exists a value of X such that (7.11) follows from (7.12). This can be done
analogously to the proof of [21, Proposition 4.1]. The result is
Proposition 7.12. We can choose X = cfx and 8 = c8X such that for every xx, x2
subject to (7.4) the following statements hold:
(i) Ifx2 = 0, then it followsfrom
(7.13)
|tj(1)(*i)|<1
and
\r}(2)(xx)\ < (2/w)Dl/4N(a)l/2/f
that
(7.14)
|*)(1)(*i)|2 < 1 - 8 and \t)(2)(xx)\* a \ïï2)(x2) |V/2 - «•
Conversely,(7.14) implies (7.3).
(ii) Let x2 J= 0; then (7.14) and (7.3) are equivalent.
8. Complexity Analysis. It is well known that the continued fraction algorithm
computes a fundamental unit of an order of a real quadratic field in 0(R'D'11)
binary operations (for every p' > 0), where R' is the regulator and D' is the absolute
value of the discriminants of the order, cf. [19]. An analogous result is true for
Voronoi's generalized continued fraction algorithm in complex cubic fields, cf. [19].
The purpose of this section is to prove that Algorithm 4.1 and Algorithm 5.2 are of
the same complexity.
Since by [4] the period length of the GVA in 0 is O(R), also the number of
iterations in Algorithm 4.1, step 2, is O(R). Since we use LLL-matrices to represent
the ideals a in Algorithm 4.1, it follows from Corollary 6.3 and from (7.1) that step
2(a) requires O(D^) binary operations (for every p > 0).
We must now analyze step 2(b), i.e., Procedure 7.1. Since 1 is a minimal point in
o, the number of iterations, when / = 2, must be O(logD). But then it follows from
the same arguments as used in the proof of [4, Lemma 2] that the number of
solutions of (7.3) with / = 1 is 0(1). The number of iterations of the procedure is
therefore 0(logD).
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48
JOHANNES BUCHMANN
It remains to analyze the complexity of the search for xx in Procedure 7.1. It
follows from Proposition 7.12 that, instead of searching the convex body described
by (7.3), which has irrational constraints, we can search the convex body described
by (7.4) and (7.14), which has rational constraints. Though the algorithm of Fincke
and Pohst [6] has turned out to be very efficient in practice, we cannot use the
complexity analysis provided for this algorithm in [6]. Therefore, we replace this
method by a procedure of Lenstra [12], which, however, is too complicated for
practical implementation. This procedure solves our search problem in polynomial
time in the size of the input data, because obviously, our convex set is "solvable", cf.
[12, Remark (d) in Section 2]. In view of (2.1), (2.2), Corollary 6.3, (7.1), (7.5), and
the choice of X in Proposition 7.12, the input data length is 0(logD). Hence,
Procedure 7.1 requires 0(0^) binary operations (for every p > 0).
Finally, it follows from (7.1) and the fact that by [4] the number of factors in step
3 of Algorithm 4.1 is O(R), that e can be computed in 0(RD>L) steps (for every
p>0).
Concluding these remarks, we have
Theorem 8.1. A fundamental unit of 0 can be computed by means of Algorithm 4.1
in 0(RD>i) binary operations (for every p > 0).
The same complexity can be proved for Algorithm 5.2.
4,-
9. Example. Let 0 = Z[p], p = v —326 . Since condition (5.1) is satisfied, we can
apply Algorithm 5.2:
1. Initialization: k <- 0, R «- 0, tj0 «- 1, a *- 0.
2. (a)
(b)
a <- 0.
tj = 18 + 6p + p2 is up to association the only number such that t) is the
2-neighbor of 1 in n, tj, «- tj.
(c) R *- 7.8633.
(d)
a, <- 1, a2 «- p, a3 <- p2/2, a4 «- p3/2, (l/iJl)a
(e) R <- 14.3402.
(0 a((l/T,,)Q) = (1/Vi)a, e «- ihA(li)
= 04
Za, * o(a).
= 1 + 108p - 36p2 + 6p3.
10. Numerical Results. We have computed the GVA-expansions in the orders
0 = Z[f^d], d e N, d * 4k4 for 1 < d < 500. In Table 1,
Ex + E2p + E3o2 + £>3
4,-
is a fundamental unit of 0, p = v - d.
In Table 2 we denote by
PL
REG
the period length,
the regulator,
NR
the relative norm of the fundamental units over Z[/—~d ].
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49
FUNDAMENTAL UNIT OF TOTALLY COMPLEX QUARTIC ORDERS
Table 1
Fundamental units of z[( -d)l/A], 1 < d < 40.
D
Ex
i
o
2
3
5
6
7
8
9
10
11
12
-1
2
-2
1
36
1
-485
-27
-98
23
-1
1
1
-1
-4
9
-2
0
-1
-45
4
3
2
-7
0
-1
1
0
-2
2
4
-26
2
198
12
96
-14
28
2
13
-86
14
-13
4
15
16
-204
16
577
7
17
-16
58332
18
-13823
11821890
19
14439374
6
20
-9
21
1
36
440140
22
-91167
-8909082
23
25899588
46
24
-95
1405
25
-4443
12
26
-125
220224
27
-1414178
2
28
57
-411912
29
330206
57396
30
-74879
28184
31
1975104
-360
32
6913
-74
33
67
-4
34
-33
-77790
35
64926
-390
36
1351
289258
37
-571878
-183064217333724
38 -267790150327167
-2544
39
3745
-94
40
-159
-36792
-11320425
-2
-24
-267972
352849
-10
0
17
135531
-12
189709
-21012
-371631
-1008
32
8
34255
0
-71847
147714465189436
840
78
1
-1
0
0
2
2
1
-66
-3
6
1
-10
-2
4
51
-1
11512
4956000
0
8
81146
1629810
-3
-281
-13
-126466
7
-39122
2218
217672
663
-6
-4
-6768
65
-6356
-54441051012990
-68
-29
Table 2
Regulators of z[(-d)1/4
D
PL NR
l
2
3
5
6
7
8
9
10
2
3
5
1
2
8
2
12
13
16
6
11
12
13
14
7
2
REG
D
PL NR
1.7627
2.4485
3.3258
3.5796
5.9660
8.9161
4.8969
13.7546
7.9923
11.7560
7.9666
10.3107
15
16
17
18
19
20
21
22
23
24
25
26
27
4
10
2
20
38
2
6
26
38
6
13
9
36
6.8013
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
1
1
1
1
1
1
1
1
1
1
-1
-1
1
REG
8.2853
14.1020
6.9942
25.3105
36.8971
6.4677
10.7870
29.4799
35.5300
10.7298
18.1845
11.4356
29.9319
50
JOHANNES BUCHMANN
Table 2 ( continued )
D
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
PL
8
23
36
28
10
4
4
28
14
27
70
16
10
9
14
72
58
28
6
62
8
40
35
26
16
21
104
10
4
12
11
120
4
19
4
2
1
2
104
2
40
36
64
2
14
9
48
22
10
44
116
4
52
15
112
20
33
126
8
NR
1
-1
1
1
1
1
1
1
1
-1
1
1
1
-1
1
1
1
1
1
1
1
1
-1
1
1
-1
1
1
1
1
-1
1
1
-1
1
1
-1
1
1
1
1
1
1
1
1
-1
1
1
1
1
1
1
1
-1
1
1
-1
1
1
REG
10.
29.
25.
31.
19.
11.
9.
25.
15.
2793
1595
0462
1193
5876
9390
4774
9225
8035
3596
3594
9285
8874
5768
28.
70.
18.
13.
11.
20. 9260
65. 4578
44. 6300
30. 5438
10. 6957
65. 8973
13. 3031
47. 6012
38. 7416
29. 8215
19. 2904
24. 6862
96. 5684
19. 9494
8. 1859
21. 0419
14. 3255
120. 9619
9. 6399
25. 0618
11. 0587
6. 2305
6. 2461
11. 1212
114. 9532
9. 7650
42. 9765
43. 1795
75. 4121
8. 4368
26. 4517
13. 7289
54. 9405
34. 8990
14. 0040
53. 3431
143. 0598
14.
63.
17.
100.
20.
33.
103.
13.
3186
4589
2367
7354
6834
3199
5648
9061
D
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
PL NR
40
44
22
34
28
56
16
32
18
30
2
26
18
13
76
186
60
8
53
106
68
27
64
10
26
12
78
116
38
28
136
126
20
22
97
10
16
9
14
166
34
72
43
96
20
76
344
16
2
3
16
272
54
2
88
68
32
3
22
REG
51
FUNDAMENTAL UNIT OF TOTALLY COMPLEX QUARTIC ORDERS
Table 2 ( continued )
D
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
PL
38
42
189
34
40
42
34
36
4
2
105
18
140
56
10
168
248
36
46
224
258
24
39
57
30
12
165
188
12
14
6
40
430
22
71
48
46
52
4
108
134
8
50
26
164
44
50
108
40
62
155
12
170
76
38
147
38
86
56
NR
REG
D
54 8322
1209
0644
206
207
208
209
210
55
194
52
53
51
56.
51.
13.
8,
98.
29,
140.
63.
17.
176.
255,
42.
45.
226.
262.
28.
60.
57.
45,
29.
151.
195,
23,
23,
14,
56,
436,
31,
93,
42.
7798
3334
3585
7027
5545
8075
5164
3062
2540
2070
9380
0583
2886
1354
4639
3833
4442
0549
3465
1234
0557
0703
3731
4328
8745
1631
5120
8157
5423
3769
3516
3738
8558
6027
7444
7772
2256
8083
1458
6175
3695
6173
7329
7403
2099
7549
5225
6433
4730
9320
4831
7578
4960
59,
51,
11,
114,
147,
20,
60.
43
190,
63,
71,
118,
49,
77,
128.
15.
162.
77.
47.
151,
38. 4180
89. 9429
46. 7214
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
PL NR
REG
44
8
50 .1061
12 .2720
14
120
74
184
86
40
250
158
48
8
53
16
20
20
12
256
12
68
2
292
30
7
114
8
70
92
14
154
106
188
48
88
8
50
322
270
134
206
20 .6215
116 .8674
74 .1316
204 .1566
85 .6995
55 .5989
252 .4171
179 .6845
53 .6943
15 .4263
61 .8690
23 .9217
25 .9591
31 .5731
29 .2041
287 .3029
27 .2051
92 .6691
12 .2875
300 .2000
37 .5111
18 .6889
137 .2790
20 .7683
82 .9100
94 .4689
25 .1475
162 .1329
120 .1679
161 .0306
45 .0959
95 .7492
16 .5705
67 .9555
355 .5986
269 .3875
145 .6830
114
174
22
116
159
318
32
34
4
70
90
13
74
96
12
28
360
94
10
194 .9075
107 .7667
177 .9984
32 .0201
142 .7931
173 .5928
334 .1780
41 .5299
49 .4254
13 .2797
78 .7884
112 .8158
19 .5379
83 .6537
89 .4168
23 .0330
33 .4639
388 .5790
106 .1759
18 .1441
52
JOHANNES BUCHMANN
Table 2 ( continued )
D
PL
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
46
42
174
50
273
62
128
40
28
282
283
284
285
286
287
288
289
290
291
292
293
294
295
4
90
8
217
314
38
40
38
64
498
128
4
116
150
14
118
147
18
46
271
288
16
296
86
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
60
83
86
8
26
216
232
134
51
18
482
12
370
62
94
4
61
11
20
8
79
4
58
4
4
4
2
NR
REG
53, 4454
52. 5164
171, 0466
65. 1967
273. 0119
68. 0688
108, 5475
55, 9533
45. 2243
13, 9617
100. 3821
15. 6401
226. 3740
355, 4098
46. 5751
46. 5511
53 5975
78, 7796
534. 2976
160, 7741
13, 3667
136. 5902
165. 4356
25. 3105
144. 4419
151. 2632
26. 7751
52. 1155
217. 7830
314. 4971
33. 6647
95, 7995
81. 3816
71. 4501
92, 5525
18, 5757
36, 0290
218, 2099
235. 1681
147. 5885
67. 4837
29. 6483
490. 9257
20. 2496
368. 9610
64. 4781
120. 4224
10. 7129
87. 0739
18. 2544
33. 8935
23. 0731
80. 5413
12. 1182
72. 4672
12. 9354
13. 5139
14. 3279
7. 8586
D
PL
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
1
2
2
2
222
2
166
4
10
126
240
2
16
9
270
10
190
6
454
76
56
49
122
4
257
352
80
78
148
56
52
144
616
28
6
120
215
66
182
73
134
470
114
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
144
46
66
189
36
10
168
168
232
12
57
38
820
24
218
215
502
NR
REG
7 .8617
14 .3402
13 .5324
12 .9601
222 .7142
12 .1552
200 .8754
23 .1719
34 .0602
154 .5054
272 .3563
10 .7870
36 .3692
17 .7100
307 .0294
20 .4472
194 .7969
29 .9819
436 .8904
78 .8263
59 .2368
61 .1623
160 .9133
18 .8724
304 .3413
152 .6403
82 .8841
82 .0567
80 .3032
179 .1556
70 .2960
53 .3035
123 .0712
646 .7992
35 .6185
25 .9792
163 .5475
222 .2816
75 .3363
191 .6409
100 .3562
148 .8015
420 .3721
142 .1201
103 .1073
181 .5445
38 .7761
16 .2358
166 .4816
161 .2338
261 .0032
26 .1178
59 .7611
71 .0890
865 .4943
30 .6399
223 .3955
256 .2240
524 .9965
FUNDAMENTAL UNIT OF TOTALLYCOMPLEXQUARTIC ORDERS
53
Table 2 ( continued )
D
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
PL NR
74
72
14
36
80
91
32
162
64
64
493
46
24
260
2
28
108
80
58
208
160
236
44
60
42
223
68
64
142
334
206
324
82
32
124
104
38
18
562
34
176
50
116
232
422
14
28
468
38
168
8
128
70
376
64
434
40
40
58
1
1
1
1
1
-1
1
1
1
1
-1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
-1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
REG
D
PL
85 .8386
76 .2725
19 .7568
42 .9218
68 .9839
88 .9151
33 .6724
171 .7357
77 .3254
89 .0531
541 .8796
43 .7689
44 .2289
248. .2283
13..9520
41. .2984
145. .4757
99..6167
66. .0425
265. .5586
215. .3186
257. .7341
43. .1812
59..3021
58..9399
275. .1346
81. 7698
76. 6110
152. 9192
288. 8929
223. 2892
306. 4599
91. 4846
46. 7187
151. 9582
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
158
50
16
22
52
68
50
34
144
8
92.
41.
23.
566.
38.
193.
66.
133.
256.
401.
30.
478
479
480
35.
436.
59.
175.
25.
131.
103.
329.
69.
388.
68.
52.
69.
2576
1450
1981
4572
9541
3631
1080
3298
8214
2819
5790
7034
4574
8639
7602
3625
6412
3975
6845
6490
0275
4719
1058
9187
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
308
1062
188
12
41
206
448
156
71
126
778
42
6
24
428
30
226
352
82
164
148
72
66
10
284
14
364
86
51
88
78
108
161
294
184
174
264
220
566
188
203
244
46
50
6
154
46
140
NR
REG
185 .2916
54 .7087
34 .7171
38 .3673
62 .8728
82 .2346
88 .9369
37 .2955
158 .3658
22 .1945
266 .8515
1211 .1332
187 .0916
21 .6822
50 .0053
229 .7405
491 .2462
159 .9725
73 .7591
114 .3615
766 .1231
58 .3189
16 .5710
31 .8975
471 .6362
36 .9668
274 .8130
371 .6888
83 .2163
190 .5559
176 .9863
94 .1183
100 .7383
22 .9074
294 .5090
25 .8042
361 .1309
100 .1847
65 .8861
116 .3825
76 .4443
119 .7289
183 .4538
311 .9160
185 .0248
156 .7782
277 .8649
274 .5838
559 .1352
190 .6584
197 .4414
286 .6531
66 .3733
62 .2386
19 .6465
204 .6873
79 .0547
161 .6921
54
JOHANNES BUCHMANN
Notice that NR = -1 if and only if PL is odd for 5 < d < 500.
The computations were carried out on the CDC-Cyber 76 of the Universität zu
Köln and the VAX 11/785 of the Department of Electrical Engineeringof The Ohio
State University.
Department of Mathematics
The Ohio State University
Columbus, Ohio 43210
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