Perfect forms over totally real number fields
aa r X i v : . [ m a t h . N T ] A ug PERFECT FORMS OVER TOTALLY REAL NUMBER FIELDS
PAUL E. GUNNELLS AND DAN YASAKI
Abstract.
A rational positive-definite quadratic form is perfect if it can bereconstructed from the knowledge of its minimal nonzero value m and the finiteset of integral vectors v such that f ( v ) = m . This concept was introduced byVorono¨ı and later generalized by Koecher to arbitrary number fields. Oneknows that up to a natural “change of variables” equivalence, there are onlyfinitely many perfect forms, and given an initial perfect form one knows howto explicitly compute all perfect forms up to equivalence. In this paper weinvestigate perfect forms over totally real number fields. Our main resultexplains how to find an initial perfect form for any such field. We also computethe inequivalent binary perfect forms over real quadratic fields Q ( √ d ) with d ≤ Introduction
Let f be a positive-definite rational quadratic form in n variables. Let m ( f ) bethe minimal nonzero value attained by f on Z n , and let M ( f ) be the set of vectors v such that f ( v ) = m ( f ). Vorono¨ı defined f to be perfect if f is reconstructible fromthe knowledge of m ( f ) and M ( f ) [Vor08]. Vorono¨ı’s theory was later extendedby Koecher to a much more general setting that includes quadratic forms overarbitrary number fields F [Koe60]. Koecher also generalized a fundamental resultof Vorono¨ı, which says that modulo a natural GL n ( O )-equivalence, where O is thering of integers of F , there are only finitely many n -ary perfect forms. Moreover,there is an explicit algorithm to determine the set of inequivalent perfect forms,given the input of an initial perfect form [Vor08, Gun99].Vorono¨ı proved that the quadratic form A n is perfect for all n , and using this wasable to classify n -ary rational perfect forms for n ≤
5. In this paper, we considertotally real fields F and explain how to construct an initial perfect form. Ratherthan trying to give a closed form expression of such a form, we show how to use thegeometry of symmetric spaces and modular symbols to find an initial perfect form.A key role is played by the notion of lattices of E -type [Kit93]. For F real quadraticand n = 2, we carry out our construction explicitly to compute all inequivalentbinary perfect forms for F = Q ( √ d ), d ≤
66. These results complement work ofLeibak and Ong [Lei08, Lei05, Ong86].Our main interest in Vorono¨ı and Koecher’s results is that they provide topo-logical models for computing the cohomology of subgroups of GL n ( O ), where F is Date : August 18, 2009.1991
Mathematics Subject Classification.
Key words and phrases.
Perfect forms, modular symbols, well-rounded retract, Eisensteincocycle.The first named author wishes to thank the National Science Foundation for support of thisresearch through NSF grant DMS-0801214. any number field. This cohomology gives a concrete realization of certain automor-phic forms that conjecturally have deep connections with arithmetic geometry. Thecohomology of subgroups of GL n ( Z ), for instance, has a (well known) relationshipwith holomorphic modular forms when n = 2, and for higher n has connections with K -theory, multiple zeta values, and Galois representations [AM92,EVGS02,Gon01].The cohomology of subgroups of GL ( O ), when F is totally real, is related to Hilbertmodular forms [Fre90]. However, computing these models for any example, a pre-requisite for using them to explicitly compute cohomology, is a nontrivial problemas soon as O 6 = Z or n >
2. Our work is a first step towards more cohomologycomputations for F totally real, computations that we plan to pursue in the future. Acknowledgements.
We thank Avner Ash, Farshid Hajir, and Mark McConnellfor helpful conversations and their interest in our work.2.
Preliminaries
Let F be a totally real number field of degree m with ring of integers O . Let ι = ( ι , . . . , ι m ) denote the m embeddings F → R . For z ∈ F , let z k denote ι k ( z ).We extend this notation to other F -objects. For example, if A = (cid:2) a ij (cid:3) is a matrixwith entries in F , then A k denotes the real matrix A k = (cid:2) a kij (cid:3) . An element z ∈ F is called totally positive if z k > k . We write z ≫ z is totally positive.2.1. n -ary quadratic forms over F . An n -ary quadratic form over F is a map f : O n → Q of the form(1) f ( x , . . . , x n ) = Tr F/ Q (cid:18) X ≤ i,j ≤ n a ij x i x j (cid:19) , where a ij ∈ F .Our main object of study will be positive-definite n -ary quadratic forms over F .Specifically, since O n ≃ Z nm , f can be viewed as a quadratic form on Z nm . Werequire that under this identification f is a positive-definite quadratic form on Z nm .Equivalently, we can use the m embeddings ι : F → R m to view f as a m -tuple( f , . . . , f m ) of real quadratic forms of Z n . It is easy to check that a quadraticform over F is positive-definite if and only if each f i is positive-definite.2.2. Minimal vectors.
There is a value m ( f ) associated to each positive-definitequadratic form f , called the minimum of f , given by m ( f ) = min v ∈O n r { } f ( v ) . Definition 2.1.
A vector v ∈ O n r { } is a minimal vector for f if f ( v ) = m ( f ).The set of minimal vectors is denoted M ( f ).Note that v is a minimal vector for f if and only if − v is as well. In our consider-ations the distinction between v and − v will be irrelevant, and so we abuse notationand let M ( f ) denote a set of representatives for the minimal vectors modulo {± } .2.3. Perfect forms.
For most quadratic forms, knowledge of the set M ( f ) is notenough to reconstruct f . A simple example is provided by the one-parameter familyof rational quadratic forms f λ ( x, y ) = x + λxy + y , λ ∈ ( − , ∩ Q , ERFECT FORMS OVER TOTALLY REAL NUMBER FIELDS 3 all of which are easily seen to satisfy M ( f λ ) = { e , e } , where the e i are the standardbasis vectors of Z . On the other hand the rational binary form g ( x, y ) = x + xy + y is reconstructible from the data of { M ( g ) , m ( g ) } (cf. § (cid:8) { e , e , e − e } , (cid:9) . We formalize this notion, due to Vorono¨ı for rational quadratic forms, with thefollowing definition:
Definition 2.2 ([Koe60, § . A positive-definite quadratic form f over F issaid to be perfect if f is uniquely determined by its minimum value m ( f ) and itsminimal vectors M ( f ). That is, given the data { M ( f ) , m ( f ) } , the system of linearequations(2) (cid:8) Tr F/ Q ( v t Xv ) = m ( f ) (cid:9) v ∈ M ( f ) has a unique solution.We warn the reader that there are other notions of perfection for quadratic formsover number fields in the literature, notably in the work of Icaza [Ica97] and Coulan-geon [Cou01]. All notions involve the reconstruction of f from its minimal vectors,but these authors use the norm where we have used the trace in the evaluation (1)of a form on a vector in O n . Moreover, Coulangeon uses a larger group to defineequivalence of forms. 3. Positive lattices
Lattices of E -type. For these results we follow [Kit93].
Definition 3.1.
Let
V / Q be a vector space with positive-definite quadratic form φ . A lattice L ⊂ V is a positive lattice for φ if for a Z -basis B = { e , . . . , e n } of L ,the associated symmetric matrix for φ in B -coordinates has rational entries.We will denote a positive lattice for φ by pair ( L, φ ). As before, one can defineminimal vectors and the minimum for a positive lattice (
L, φ ). We denote these M ( φ ) and m ( φ ), with L understood.Given two positive lattices ( L , φ ) and ( L , φ ), with L ⊂ V and L ⊂ V , onecan construct a new positive lattice ( L ⊗ L , φ ⊗ φ ). Specifically, let B denotethe symmetric bilinear form giving rise to φ , and let B denote the symmetricbilinear form giving rise to φ . We define a symmetric bilinear form B on V ⊗ V by first defining B ( v ⊗ w, ˆ v ⊗ ˆ w ) = B ( v, ˆ v ) B ( w, ˆ w )on simple tensors and then by linearly extending to all of V ⊗ V . Then one has apositive-definite quadratic form φ = φ ⊗ φ on V ⊗ V given by φ ( x ) = B ( x, x ).Note that by construction, we have φ ( v ⊗ w ) = φ ( v ) φ ( w ). Definition 3.2.
A positive lattice (
L, φ ) is of E -type if M ( φ ⊗ φ ′ ) ⊂ { u ⊗ v | u ∈ L, v ∈ L ′ } for every positive lattice ( L ′ , φ ′ ). PAUL E. GUNNELLS AND DAN YASAKI
In other words, a lattice is E -type if, whenever it is tensored with another positivelattice, the minimal vectors of the tensor product decompose as simple tensors.Positive lattices of E -type are particularly well-behaved with respect to tensorproduct: Proposition 3.3 ([Kit93, Lemma 7.1.1]) . Let ( L, φ ) and ( L ′ , φ ′ ) be positive lattices.If ( L, φ ) is of E -type, then (i) m ( φ ⊗ φ ′ ) = m ( φ ) m ( φ ′ ) , and (ii) M ( φ ⊗ φ ′ ) = { x ⊗ y | x ∈ M ( φ ) , y ∈ M ( φ ′ ) } . The form A n . We now give an example that will be important in the sequel.Let A n be the rational quadratic form A n ( x , . . . , x n ) = X ≤ i ≤ j ≤ n x i x j . It is easy to check, as was first done by Vorono¨ı [Vor08], that A n is perfect. Onecomputes m ( A n ) = 1 and M ( A n ) = { e i } ∪ { e i − e j } . According to [Kit93, Theorem 7.1.2], ( Z n , A n ) is of E -type.3.3. Application to n -ary quadratic forms over F . Fix α ∈ F totally positive.Consider the n -ary quadratic form over F given by(3) f α ( x , . . . x n ) = Tr F/ Q ( αA n ( x , . . . , x n )) . Lemma 3.4.
We have ( O n , f α ) = ( O ⊗ Z n , φ α ⊗ A n ) , where φ α ( x ) = Tr F/ Q ( αx ) .Proof. It is clear that as Z -modules, we have O n ≃ O ⊗ Z n . Thus we want to showthat under this isomorphism, the quadratic form f α on O n is taken to the quadraticform φ α ⊗ A n on O ⊗ Z n . We do this by explicit computation. Let a ∈ O and let x = P x i e i ∈ Z n . Then we have( φ α ⊗ A n )( a ⊗ x ) = φ α ( a ) A n ( x )= Tr F/ Q (cid:0) αa (cid:1) X ≤ i ≤ j ≤ n x i x j = Tr F/ Q (cid:16) α X ≤ i ≤ j ≤ n a x i x j (cid:17) = Tr F/ Q ( αA n ( a x ))= f α ( a x ) , which completes the proof. (cid:3) Theorem 3.5.
Let f α be as in (3) . Then there exist nonzero η , . . . , η r ∈ O suchthat (i) the minimum of f α is m ( f α ) = m ( φ α ) = Tr F/ Q ( αη i ) , and ERFECT FORMS OVER TOTALLY REAL NUMBER FIELDS 5 (ii) the minimal vectors of f α are M ( f α ) = [ ≤ k ≤ r { η k e i } ∪ { η k ( e i − e j ) } . Proof.
Since ( Z n , A n ) is of E -type, the result follows from Proposition 3.3 andLemma 3.4 by taking { η i } to be the minimal vectors for φ α . (cid:3) The geodesic action, the well-rounded retract, and theEisenstein cocycle
In this section we present three tools that play a key role in the proof of Theorem5.3. The geodesic action [BS73] is an action of certain tori on locally symmetricspaces. The well-rounded retract [Ash84] is a deformation retract of certain locallysymmetric spaces. The
Eisenstein cocycle [Scz93, GS03] is a cohomology class forSL m ( Z ) that gives a cohomological interpretation of special values of the partialzeta functions of totally real number fields of degree m .4.1. Geodesic action.
Let G be a semisimple connected Lie group, let K bea maximal compact subgroup, and let X be the symmetric space G/K . Fix abasepoint x ∈ X . This choice of basepoint determines a Cartan involution θ x . Fora parabolic subgroup P ⊂ G , the Levi quotient is L P = P/N P , where N P is theunipotent radical of P . Let A P denote the (real points of) the maximal Q -splittorus in the center of L P , and let A P,x denote the unique lift of A P to P that isstable under the Cartan involution θ x .Since P acts transitively on X , every point z ∈ X can be written as z = p · x forsome p ∈ P . Then Borel–Serre define the geodesic action of A P on X by a ◦ z = ( p e a ) · x, where e a is the lift of a to A P,x . This action is independent of the choice of basepoint x , justifying the notation. Note that at the basepoint x , the geodesic action of A P agrees with the ordinary action of its lift A P,x .4.2.
Well-rounded retract.
Now let G = SL m ( R ), K = SO( m ). The space X is naturally isomorphic to the space of m -ary positive-definite real quadratic formsmodulo homotheties. Indeed, this follows easily from the Cholesky decompositionfrom computational linear algebra: if S is a symmetric positive-definite matrix ofdeterminant 1, then there exists a matrix g ∈ G such that gg t = S .Let W ⊂ X be the subset consisting of all forms whose minimal vectors span R m . Then Ash proved that W is an SL m ( Z )-equivariant deformation retract of X . Moreover W naturally has the structure of a cell complex with polytopal cells,and SL m ( Z ), and thus any finite-index subgroup Γ ⊂ SL m ( Z ), act cellularly on W with finitely many orbits. The retract can be used to compute the cohomology ofΓ for certain Z Γ-modules in the following way. Let M be a Z Γ-module attached toa rational representation of SL m ( Q ) and let f M be the associated local coefficientsystem on Γ \ X . Then we have isomorphisms H ∗ (Γ; M ) ≃ H ∗ (Γ \ X ; f M ) ≃ H ∗ (Γ \ W ; f M ) . PAUL E. GUNNELLS AND DAN YASAKI
Eisenstein cocycle.
As before let O be the ring of integers in a totally realnumber field of degree m . Let b , f ⊂ O be relatively prime ideals. The partialzeta function ζ ( b , f ; s ) attached to the ray class b (mod f ) is defined by the analyticcontinuation of the Dirichlet series ζ ( b , f ; s ) = X a N( a ) − s , Re( s ) > , where the sum is taken over all integral ideals a such that ab − is principal with atotally positive generator in the coset 1 + fb − . By the theorem of Klingen–Siegel[Sie70], the special values ζ ( b , f ; 1 − k ), where k ∈ Z > , are rational.The special values have a cohomological interpretation. Let U be the group oftotally positive units in the coset 1 + f . Sczech constructed a sequence of rationalcocycles η ( b , f , k ) ∈ H m − ( U ; Q ) which give the numbers ζ ( b , f ; 1 − k ) by evaluationon the fundamental cycle in H m − ( U ; Z ). To construct the cocycles η ( b , f , k ), onespecializes a “universal” cocycle Ψ ∈ H m − (SL m ( Z ); M ), where M is a certainmodule. After choosing b , f , k , one plugs U -invariant parameters into M to obtaina module M k , which is C with a nontrivial SL m ( Z )-action, and a class Ψ( b , f , k ) ∈ H m − (SL m ( Z ); M k ). Then η ( b , f , k ) is obtained by restriction, after realizing U asa subgroup of SL m ( Z ) via a regular representation; note that M k restricted to U istrivial. 5. Scaled trace forms and the main result
We now return to perfect forms. The quadratic form φ α from Lemma 3.4 willplay an imporant role in Theorem 5.3, so we give it a name: Definition 5.1.
For α ∈ F , the scaled trace form associated to α is the map φ α : O → Q given by φ α ( η ) = Tr F/ Q ( αη ).For the remainder of the paper, fix a Z -basis B = { ω , . . . , ω m } of O . Then for x = P x i ω i ∈ O with x i ∈ Z , we have φ α ( x ) = X ≤ i,j ≤ m Tr F/ Q ( αω i ω j ) x i x j . In particular, fixing B allows us to view the form φ α as an m -ary quadratic form[ φ α ] B over Q .Let V denote the m ( m + 1) / R -vector space of m -ary quadraticforms over R . Note that for α, β ∈ F , we have[ φ α + β ] B = [ φ α ] B + [ φ β ] B . In particular the image of F ⊗ R in V is an m -dimensional subspace. The form φ α is positive-definite if α ≫
0. Let C ⊆ V denote the real cone of positive-definite m -ary quadratic forms, and let C + B denote the subcone corresponding to the totallypositive scaled trace forms. More precisely, let C + B = Cone( { [ φ α ] B | α ≫ } ) ⊗ R . Let X be the global symmetric space G/K , where G = SL m ( R ) and K = SO( m ).Recall ( § X ≃ C/ R > , where R > acts on C byhomotheties. We denote by X + B the image of C + B in X under the projection C → X . ERFECT FORMS OVER TOTALLY REAL NUMBER FIELDS 7
Lemma 5.2.
There exists an R -split torus A ⊂ G and a point x B ∈ X such that X + B = { a ◦ x B | a ∈ A } , where ◦ is the geodesic action of A on X ( § We begin with some computations in C . For α ∈ F totally positive, let S ( α ) = (cid:2) S ( α ) ij (cid:3) denote the positive-definite symmetric m × m matrix correspondingto [ φ α ] B . Thus S ( α ) ij = Tr F/ Q ( αω i ω j ) = X ≤ k ≤ m α k ω ki ω kj . We can write S ( α ) as g ( α ) g ( α ) t , where g ( α ) is given by g ( α ) ij = √ α j ω ji . This implies g ( α ) = Ω a , where(4) Ω ij = ω ji and a = diag( √ α , . . . , √ α m ) . From these considerations it is clear that the cone C + B ⊂ C is given by the set ofmatrices of the form Ω a , where a is allowed to vary over all positive real diagonalmatrices, not just those of the special form on the right of (4).Now we pass to X by modding out by homotheties. Let x B be the image of thepoint [ φ ] B . Let Υ be the unique positive multiple of Ω such that det(Υ) = 1; thisalso maps onto x B . Then the subset of C given by { Υ a | a = diag( a , . . . , a m ) , a k ∈ R > , a · · · a m = 1 } maps diffeomorphically onto X + B ⊂ X . Let P ∞ ⊂ G be the parabolic subgroupof upper-triangular matrices, and let A ∞ ⊂ P ∞ be the diagonal subgroup. Let x denote the point of X fixed by K . Note that X + B is precisely a translate of thesubmanifold defined by the geodesic action of A ∞ on x : X + B = Υ · { a ◦ x | a ∈ A ∞ } . By “transport de structure” we can express this at the basepoint, that is x B = Υ · xX B = { b ◦ x B | b ∈ Υ A ∞ Υ − } . More precisely,(5) (Υ a ) · x = (Υ a Υ − )Υ · x = (Υ a Υ − ) · x B . Now Υ a Υ − ∈ Υ A ∞ Υ − , and (5) is exactly the geodesic action of the elementΥ a Υ − on the point x B . Thus we may take A = Υ A ∞ Υ − , and the lemma follows. (cid:3) By Theorem 3.5, to find a perfect n -ary form over F , one can look for scaledtrace forms with many linearly independent minimal vectors. Specifically, a scaledtrace form f α = φ α ⊗ A n is perfect if(6) (cid:8) Tr F/ Q ( v t Xv ) = m ( f α ) (cid:9) v ∈ M ( f α ) defines mn ( n +1) / A n is perfect over Q , the minimal vectors of A n define n ( n + 1) / φ α has m linearly independentminimal vectors, then (6) will impose mn ( n + 1) / F , and f α will be perfect. We now prove ourmain result, which asserts that such an α can always be found: PAUL E. GUNNELLS AND DAN YASAKI
Theorem 5.3.
There exists α ∈ F totally positive such that φ α has m linearlyindependent minimal vectors, and thus f α ( x , . . . , x n ) = Tr F/ Q ( αA n ( x , . . . , x n )) is a perfect form over F .Proof. We must show that we can find α ≫ φ α ] B is well-rounded. ByLemma 5.2, we know that a choice of basis B gives rise to a point x B and a maximal R -split torus A such that X + B = { a ◦ x B | a ∈ A } . We will show that X + B ∩ W = ∅ ,where W is the retract for Γ = SL m ( Z ) ( § C ofany point in X + B ∩ W is a ray containing an F -rational point φ α . This will provethe theorem.To show X + B ∩ W = ∅ we use the Eisenstein cocycle ( § f = O , sothat ζ ( b , f ; s ) is the Dedekind zeta function ζ F ( s ), and U is the group of totallypositive units. Abbreviate Ψ( b , f , k ) (respectively, η ( b , f , k )) to Ψ( k ) (resp., η ( k )).Using the regular representation attached to the basis B , we have an injection i : U → Γ. Let M ′ k be the module dual to M k . Since M k and M ′ k are trivial afterrestriction to U , we obtain induced maps i ∗ : H m − ( U ; C ) → H m − (Γ; M ′ k ) and i ∗ : H m − (Γ; M k ) → H m − ( U ; C ).Let h , i ⋆ the pairing between H m − and H m − , where ⋆ is either U or Γ, andwhere the target is C ≃ M k ⊗ Γ M ′ k . Let ξ ∈ H m − ( U ; C ) be the fundamental class.Then h i ∗ ( ξ ) , Ψ( k ) i Γ = h ξ, i ∗ (Ψ( k )) i U = h ξ, η ( k ) i U = ζ F (1 − k ) . Since the special values ζ F (1 − k ) do not vanish identically, the class i ∗ ( ξ ) pairsnontrivially with Ψ( k ) for some k . But it is easy to see that i ∗ ( ξ ) is the same as theclass of X + B (mod Γ) in the homology of the quotient Γ \ X . If X + B ∩ W were empty,then by the discussion in § X + B (mod Γ) would pair trivially withall cohomology classes for all coefficient modules, which is a contradiction. Thus X + B ∩ W = ∅ .To finish we must prove that the ray above an intersection point in X + B ∩ W contains a form φ α with α ∈ F . This follows easily since W is cut out by linearequations with Q -coefficients and from the explicit form of the cone C + B . Thiscompletes the proof of the theorem. (cid:3) Real quadratic fields
Preliminaries.
Let d be a square-free positive integer, and O be the ringof integers in the real quadratic field F = Q ( √ d ). Then O is a Z -lattice in R ,generated by 1 and ω , where ω = (1 + √ d ) / d ≡ ω = √ d otherwise. The discriminant D equals d if d ≡ d otherwise.6.2. Scaled trace forms.
Let C be the cone of positive-definite binary quadraticforms. Modding out by homotheties, we can identify C/ R > with the upper half-plane H . One such identification is given by(7) x + iy (cid:20) − x − x x + y (cid:21) . Fixing a Z -basis B = { , ω } for O , we consider the subcone C + B ⊂ C of totallypositive scaled trace forms as described in § ERFECT FORMS OVER TOTALLY REAL NUMBER FIELDS 9
By Lemma 5.2, it follows that the image of C + B is a geodesic in H . Considering(7) and our choice of basis, we see that C + B corresponds to (cid:26) (cid:20) Tr F/ Q ( α ) Tr F/ Q ( αω )Tr F/ Q ( αω ) Tr F/ Q ( αω ) (cid:21) (cid:12)(cid:12)(cid:12)(cid:12) α ≫ (cid:27) ⊗ R . On H this becomes the geodesic X + B defined by( x + 12 ) + y = d d ≡ x + y = d otherwise.The well-rounded retract W ⊂ H is the infinite trivalent tree shown in Figure 6.2.The crenellation comes from arcs of circles of the form ( x − n ) + y = 1, where n ∈ Z . One can compute a point x + iy of the intersection of W and the geodesiccorresponding to C + B . Let X ( n ) = n + d −
54 + 8 n if d ≡ n + d − n otherwise.Then x = min n ∈ Z | X ( n ) | , and y can be explicitly computed from x . Specifically,let e n be a non-negative integer such that X ( e n ) = x . Then y satisfies( x − e n ) + y = 1 . The corresponding scaled trace form is φ α , where α = d − (2 x + 1) √ d d if d ≡ d − x √ d d otherwise. Figure 1.
The well-rounded retract W ⊂ H .We summarize the results of this computation below. Proposition 6.1.
Let α and e n be as above. Let η = e n + ω . Then the scaled traceform φ α is minimized at {± , ± η } . Binary perfect quadratic forms.
Given a binary perfect form as an initialinput, there is an algorithm to compute the GL ( O )-equivalence classes of binaryperfect forms over F [Gun99]. This was done in [Ong86] for F = Q ( √ Q ( √ Q ( √
5) and in [Lei05] for F = Q ( √ ( O )-equivalence classes of these forms over F = Q ( √ d ) for square-free d ≤ Magma [BCP97] and
PORTA [CL]. The numberof GL ( O )-classes of perfect forms is given in Table 1. Figure 2 shows a plot of thedata ( D, N D ) from Table 1. D h D N D D h D N D D h D N D D h D N D Table 1. GL ( O )-classes of perfect binary quadratic forms overreal quadratic fields. The discriminant is D , the class number of Q ( √ D ) is h D , and the number of inequivalent forms is N D .40003500300025002000150010005000 25020015010050 N D D Figure 2.
Perfect forms by discriminant
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