We survey a number of classification tools developed in recent years and employ them to classify pseudo-unitary rank 5 premodular categories up to Grothendieck equivalence.

Fusion categories axiomatize and generalize the theory of representation theory, and their study encompasses the representation theory of finite groups and certain Hopf algebras. In many of these situations, the fusion category associated with the underlying group/algebra enjoys extra structure, such as a notion of commutativity (braiding), duality (rigidity), or other conditions such as a spherical structure and non-degeneracy (modularity).1 The study of fusion categories has moved beyond its roots in groups and algebras and now has a more widespread use. For instance, non-degenerate ribbon braided fusion categories, i.e., modular categories, have broad uses in physics, where they describe topological phases of matter and topological quantum computers.2,3 Modular categories also have applications in pure mathematics providing knot, link, and 3-manifold invariants through Topological Quantum Field Theories (TQFT).4 More recently, ribbon braided fusion categories, i.e., premodular categories, have garnered increased attention. These categories are thought to describe higher dimensional TQFT and thus have relevance in manifold invariants and physics.5 Furthermore, the study of premodular and modular categories often follow a “leap-frogging” pattern whereby advances in the understanding of one type of category allow for advances in another. In recent years, researchers have found it useful to stratify fusion categories by a numeric parameter known as their rank. The low-rank classification of premodular categories has historically lagged the classification of modular categories. On the premodular side, categories are understood through rank 4,6–8 while on the modular side, they are completely characterized through rank 5.9,10 Additional conditions are often placed on modular categories to make further progress on classification. For instance, weakly integral modular categories have been classified through rank 711,12 and maximally non-self dual modular categories have been classified through rank 11.11 

In this work, we aim to utilize a technique known as (de-)equivariantization to produce modular categories from premodular categories. This will allow us to leverage recent advances in arithmetic properties of modular categories13 and to classify pseudo-unitary premodular categories of rank 5 up to Grothendieck equivalence. Specifically, we will show the following:

Theorem I.1.

IfCis a pseudo-unitary rank 5 premodular category, then either

  • Cis Tannakian and is given by Rep(G), where G isZ5, D8, Q8, D14, Z5Z4, Z7Z3, S4, orA5or

  • Cis properly premodular and Grothendieck equivalent to one of the following:

    1. Rep(D14) withCRepZ2anddi=1,1,2,2,2.

    2. Rep(S4)withCRepS3, di=1,1,2,3,3, andT=diag1,1,1,1,1.

    3. Rep(D8) withCRepZ2, di=1,1,2,1,1, andT=diag1,1,θ,1,1, where θ is a root of unity satisfying a monic degree 4 polynomial overZ.

    4. SU(2)8/Z2.

  • Cis modular and it is Grothendieck equivalent toSU(2)4,SU(2)9/Z2,SU(5)1orSU(3)4/Z3.13 

Moreover, each case is realized.

In Sec. II, we will review the basic theory of premodular categories. Having dispensed with the preliminaries, we will stratify premodular categories by the amount of degeneracy (rank of the Müger center). In Sec. III, we will analyze each case in turn to arrive at Theorem III.2.

A premodular category C is a balanced, braided fusion category. We will denote the isomorphism classes of simple objects by Xa, indexed such that X0=I is the monoidal unit. We will denote the set of such isomorphism classes by Irr(C). The fusion matrices, Na, the S-matrix, S = (sx,y), and the T-matrix, T=(δx,yθx), are defined in the usual way.1 Here θx is the twist of the simple x and is known to have finite order.14 The triple ({Na}, S, T) is known as premodular datum. Throughout we will assume C is pseudo-unitary, and so we may take the categorical dimensions, da, to be the Frobenius-Perron eigenvalues of the Na, i.e., the FP-dimensions. This simplifying assumption has physical relevance in that only pseudo-unitary categories can produce physical theories.5 

Let C denote the Müger center of the category C.15 We say that C is modular if C=Vec, symmetric if C=C, and properly premodular otherwise. Symmetric categories are completely characterized by the following result due to Deligne:16 

Theorem II.1.

IfCis a symmetric category, thenCRep(G,z), where z is a central element of G of order at most 2.

In the properly premodular setting, this result can be exploited to determine part of the premodular datum. For instance, the categorical dimensions of simples in CRep(G,z) follow from the representation theory of G. Of course, by definition of the S-matrix, for isomorphism classes of simples, X and Y, in C, one has sX,Y = dXdY. One can obtain similar information regarding the T-matrix, for instance:

Lemma II.2.

SupposeCis a symmetric ribbon category andXCis an object such thatXXI, then θX = ±1. Moreover, θX = 1 if X fixes an element ofC. In particular, this is true ifCis the Müger center of an odd rank premodular category.

Proof.

The first statement in this lemma can be attributed to Deligne.16 Now if θX = −1 and XXI, then we know that ⟨1, X⟩ ≅ sVec. This means that XYY for all simples Y in the category17 [Proposition 2.6 (i)].

Remark II.3.

There are non-Tannakian odd rank fusion categories; however these categories do not admit a ribbon structure17[Proposition 2.6(i)]. In concrete cases, this can be seen directly by inspecting the balancing equation (1). For instance, consider the non-Tannakian symmetric fusion categoryRepQ8. This category has rank 5 with dimensions 1, 1, 1, 1, 2. Letting X denote the isomorphism class of the 2-dimensional simple object and f denote the isomorphism class of the generator of sVec inRepQ8. Then fX = X. On the other hand, the category is symmetric and so 2 = dX = dXdf = SX,f. The right-hand side can be evaluated from the balancing equation to giveSX,fθXθf=dXfθXf=dXθX=2θX. In particular, we have 2 = SX,f = 2/θf = −2, an impossibility.

In the case that C has odd rank, Lemma II.2 tells us that the Müger center of C is Tannakian. In particular, we can exploit Bruguières’ minimal modularization.18 This procedure is more generally known as de-equivariantization. Recent work by Natale and Burciu allows one to gain a great deal of insight into the structure of the minimal modularization of C.19 Moreover, it is often true that the structure enforced by the de-equivariantization procedure is at odds with the Galois theory of the modular category. By exploring the interplay between these two tools, one can make progress in the classification of premodular categories. A complete discussion of de-equivariantization and the Galois theory of modular categories is beyond the scope of this current work, but details can be found in many papers.12,18,19

Next, recall that the T-matrix, S-matrix, and dimensions are related through the balancing equation as follows:1 
θXθYSX,Y=k=04NX*Ykθkdk.
(1)
Furthermore, recall that dX are further related through the dimension equation as follows:
dXdY=TrC(XY)=TrC(NX,YZZ)=ZNX,YZdZ.
(2)

The structure of the fusion rules, knowledge of dimensions, and twists of a subcategory often allow one to produce a pair of polynomials from these equations from which the order of certain twists can be bounded. In particular, we have the following:

Lemma II.4.
IfDis a subset ofda1ar, fQD[x]is a degree 2 polynomial with leading coefficient a2and constant coefficient a0such thatfθ=0for some θ, then
[Qθ:Q]2D+1ifa20 and a0/a2is a unit,2Difa2=0 or a0/a2is not a unit.

Proof.
Let α be a primitive element of QD Then the minimal polynomial of θ over QD divides f in QD. In particular, if we let m denote the degree of QD,θ over QD, then m ≤ 2. Note that the minimal polynomial of θ over QD must divide the minimal polynomial of θ over Q. Since θ is a root of unity, we can conclude that the minimal polynomial over QD can only possibly have degree 2 if a2 ≠ 0 and a0/a2 is a unit. In particular,
m2if a20 and a0/a2 is a unit,1if a2=0 or a0/a2 is a unit.

Next note that for any dD, d satisfies a degree 2 monic polynomial over QD\d. In particular, QD has degree at most 2D over Q. Thus QD,θ has degree at most m2D over Q. Since Qθ is a subfield, the result follows.

Lemma II.5.

LetCbe a rank r premodular category with rank r − 1 Müger center and order the simples so that the last object, Xr−1, is not inC. ThenSr1,r1=dimC.

Proof.
Since Xr−1 is not in the Müger center, the (r − 1)-th column of the S-matrix must be orthogonal to the 0-th column. However, S,0 = d for 0 ≤ r − 1, and Sk,r−1 = dkdr−1 for 0 ≤ kr − 2. Thus the orthogonality condition reads
0=j=0r1Sj,0Sj,r1=j=0r1djSj,r1=j=0r2djdr1dj+dr1Sr1,r1.
Of course j=0r2dj2=dimC and so dr1dimC=dr1Sr1,r1. Since dr−1 ≠ 0, the result follows.

In this section, we will classify rank 5 pseudo-unitary premodular categories up to premodular datum, and in particular, Grothendieck equivalence. Our focus on pseudo-unitarity is physically motivated as pseudo-unitarity is a necessary condition for the category to lead to a physical theory.5 The classification of pseudo-unitary rank 5 modular categories was recently completed,13 so it suffices to consider symmetric and properly premodular categories.

We begin with symmetric premodular categories. By Theorem II.1 and Lemma II.2, it suffices to determine all groups which have exactly 5 irreducible representations. The number of irreducible representations can be related to the order of the group via a classical number theoretic argument11 [lemma 4.4 (ii)]. In particular, we have

5|G|A55|G|1806.

Applying these bounds, one can perform an exhaustive search in GAP and Lemma II.2 to deduce the following:

Proposition III.1.

IfCis a symmetric rank 5 category, thenCis Tannakian and is given by Rep(G), where G isZ5, D8, Q8, D14, Z5Z4, Z7Z3, S4, orA5.

Having completed the classification of rank 5 symmetric categories, we find it convenient to stratify our analysis of rank 5 properly premodular categories by the rank of the Müger center. From Propositions III.1, III.3, III.7, and III.11, we will be able to prove the following:

Theorem III.2.

IfCis a pseudo-unitary rank 5 premodular category, then one of the following holds:

  • Cis Tannakian and is given by Rep(G), where G isZ5, D8, Q8, D14, Z5Z4, Z7Z3, S4, orA5.

  • Cis properly premodular and Grothendieck equivalent to the following:

    1. Rep(D8) withCRepZ2,di=1,1,2,1,1, andT=diag1,1,θ,1,1, where θ is a root of unity satisfying a monic degree 4 polynomial overZ.

    2. Rep(D14) withCRepZ2anddi=1,1,2,2,2.

    3. Rep(S4)withCRepS3, di=1,1,2,3,3, andT=diag1,1,1,1,1.

    4. SU(2)8/Z2.

  • Cis modular and it is Grothendieck equivalent toSU(2)4,SU(2)9/Z2,SU(5)1orSU(3)4/Z3.13 

Moreover, each case is realized.

Proposition III.3.

There is no rank 5 properly premodular category with a rank 4 Müger center.

In order to prove this result, we proceed through Propositions III.4, III.5, and III.6. These propositions consider the different structures of C.

By Theorem II.1, the structure of C is Rep(G, z) as a symmetric category. Moreover, we have |G| ≤ 4211 [Lemma 4.4(ii)]. Using GAP, we can conclude that G is isomorphic to Z4, Z2×Z2, D10, or A4. In each of these cases, we may apply Lemma II.5 to deduce that s4,4 = −4. Furthermore, in each case, we can apply Theorem II.1 and Lemma II.2 along with (1) applied to Sa,a for 0 ≤ a ≤ 4 to deduce that θa = 1 for 0 ≤ a ≤ 3. In particular, the S- and T-matrices are determined by the dimension, d, and twist, θ, of X4. Moreover, by Lemma II.4, we can conclude that the order, n, of θ satisfies n1,2,3,4,5,6,8,10,12. The cases of Z4 and Z2×Z2 are sufficiently similar that we consider them in tandem.

Proposition III.4.

There is no rank 5 premodular category with Müger center Grothendieck equivalent toRepZ4orRepZ22.

Proof.

In both cases, the only n1,2,3,4,5,6,8,10,12 consistent with Eqs. (1) and (2) is n = 10; consequently, d=1+5. Letting G denote the group such that CRepG, we can de-equivariantize to produce a modular category CG. Let D^ denote the set of orbits of isomorphism classes of the G-action, and let D denote a transversal of D^. Then the simples of C can be understood via equivariantization as pairs y,V, where y runs over D and V ranges over the simple projective representations of StabGy.19 Since I is stabilized by G and G is an abelian group, four of the simples in C are of the form I,χa, where χa is a irreducible character of G. The remaining simple is ŷ,V^, where V^ is the sole simple projective representation of StabGŷ. In the case of GZ4, we recall that H2Z2k,C×=0 for all k, and so we may take V^ to be the sole irreducible linear representation. In particular, dimŷ=1+5/4Z¯, an impossibility. So it suffices to consider GZ22. In this case, H2Z22,C×Z2, and so V must be the 2-dimensional simple projective corresponding to the nontrivial 2-cocycle in H2Z22,C×. In particular, CZ22Fib. This is not possible.20 

Proposition III.5.

There is no rank 5 premodular categoryCwithC=Rep(D10).

Proof.

The only choice for n1,2,3,4,5,6,8,10,12 consistent with Eqs. (1) and (2) is n = 4 or 12. In both cases, we have d = 3 and dimC=19. This is not possible as dimC=10dimC=19.

Proposition III.6.

There is no rank 5 premodular categoryCwithC=Rep(A4).

Proof.

The only choice for n1,2,3,4,5,6,8,10,12 consistent with Eqs. (1) and (2) is n = 4 or 12. In both cases, we have d=23. Next we observe that Z3A4, and so CA4 is a premodular category with Z3 action. Letting D denote a transversal of the orbits of the simple object isomorphism classes under this action, we can conclude that the isomorphism classes of simple objects in C are described by pairs y,V, where yD and V is a simple projective representation of StabZ3y. Since y=I is fixed by the Z3-action, we can deduce that three of the simples are I,χa, where the χa are the irreducible characters of Z3. The remaining two objects we denote by y,V and z,W. Since Z3 has no proper subgroups, it must be that yz and StabZ3y=StabZ3z=1. Applying the orbit-stabilizer theorem, we know that Z3.y=Z3.z=3. Thus y has dimension 2/3Z¯, an impossibility.

Proposition III.7.

IfCis a rank 5 properly premodular category with a rank 3 Müger center, thenCis Grothendieck equivalent toRepS4, CRepS3, andT=diag1,1,1,1,1.

We will consider the different possibilities for C in Propositions III.9 and III.10. However, before doing this, we find it useful to establish the following lemma.

Lemma III.8.

SupposeCis a modular category with da = db ≠ 1 for ab and there existsσGalCsuch thatσa=0, thendσb=±1. In particular,CptVec.

Proof.

This follows from the fact that dσ0σdk=dσkϵσk,σσda=σdb9 (Theorem 2.7).

Next, notice that groups with 3 irreducible representations must satisfy 3 ≤ |G| ≤ A3 ⇒3 ≤ |G| ≤ 611 [Lemma 4.4(ii)]. In particular, GS3 or Z3.

Proposition III.9.

There is no rank 5 properly premodular category whose Müger center is Grothendieck equivalent toRepZ3.

Proof.

Suppose C is such a properly premodular category. Then C is Tannakian since Z3 has no order 2 central elements. Thus CZ3 is modular. Now let D^ be the set of orbits of isomorphism classes of simples in CZ3 under the Z3 action, and let D be a transversal of D^. Then the isomorphism classes of simples in C are given by I,χa, y,V, and z,W, where χa are the irreducible linear characters of Z3 and V and W are simple projective representations of StabZ3y and StabZ3z, respectively. Since Z3 has no proper subgroups and H2Z3,C×=0, we may take V and W to be irreducible linear representations. Of course all irreducibles of the stabilizer must appear, and so yz and the stabilizers are trivial. Applying the orbit-stabilizer theorem, we can conclude that Z3.y=Z3.z=3 and CZ3 is a rank 7 modular category of global dimension 1 + 3a2 + 3b2, where a = d3/3 and b = d4/3. So either a = b = 112 or we may apply Lemma III.8 to conclude that a = 1. Applying the equidimensionality of the universal grading of CZ3,21,22 we can deduce that b = 1 or 2. The latter case cannot occur as every integral modular category of rank 7 is pointed12 (Theorem 5.8). Thus the global dimension of C is 21. From here, we can conclude that C is Grothendieck equivalent to RepZ3Z723 (Theorem 6.3). Applying these fusion rules, balancing, and the symmetry s34 = s43 allows us to conclude that θ3 = θ4. The last column of S is now determined in terms of θ3 by balancing and the fusion rules.

Applying orthogonality of the first and last columns of the S-matrix produces the equation θ32+5θ3+1=0. This is not possible as θ3 is a root of unity.

Proposition III.10.

IfCis properly premodular andChas rank 3, thenCis Grothendieck equivalent toRepS4, CRepS3, andT=diag1,1,1,1,1. Such a category can be obtained via theS3-equivariantization of a 3-fermion theory.

Proof.

Just as in the proof of Proposition III.9, we may de-equivariantize and understand the simples of C in terms of equivariantization. Since H2S3,C×=0, we have that the isomorphism classes of simples are I,ρa, y,V, and z,W, where ρa are the irreducible representations of S3, while V and W are the irreducible linear representations of StabS3y and StabS3z, respectively. There are now two cases to consider:

Case 1:y = z. In this case, StabS3y can only have two irreducible linear representations and hence is isomorphic to Z2. By the orbit-stabilizer theorem, we can conclude that CS3 has rank 4 and thus is pointed.9 

We know that the simple objects in C have dimensions 1, 1, 2, 3, 3,19 and the balancing equation easily tell us that θ0 = θ1 = θ2 = 1. In addition, if we de-equivariantize by Rep(Z2)Rep(S3)=C, we get a premodular category with simples with dimensions 1, 1, 1, 3. From here, we can conclude that x3=x3* and x4=x4*19 (Proposition 3.21). In addition, we know that X1X3 = X4X3X3 does not contain X1 in its decomposition. If we know look at the dimension equation corresponding to X3X3, we get the following Diophantine equation:
9=1+2N332+3(N333+N334)N332=1 and N333+N334=2 or N332=4 and N333+N334=0.

The latter case cannot happen since this would force 4 ≤ ||N3||maxd3 = 310 (Lemma 3.14). From here, we can finally conclude that N233=N234=N243=N244=N134=1 and N333+N334=N443+N444=2. If we now use the fact that all the Ni matrices commute, we can conclude that N334=N444 and N443=N333. If we combine the previous mentioned observations and the equation that you get by looking at s33, we are able to conclude that θ3 = θ4 = −1.

Note that if we consider the rank 4 pointed modular categories, D, then the Grothendieck group is either Z4 or Z22. In the former case, there are two S3 equivariantizations of rank 9 and 12. In the latter case, if we take D to be equivalent to the toric code, then there are again two S3 equivariantizations of rank 9 and 12. The final option is that D is Grothendieck equivalent to the 3-fermion theory. In this case, AutbrDS3, and so the possible S3-actions are given by group homomorphisms, HomS3,S3. Once again, there are rank 9 and 12 equivariantizations. However, the identity automorphism yields an S3-action on D with 2 orbits. The stabilizer of the nontrivial orbit is Z3, and so the S3-equivariantization under such an action would have rank 5 and have simples of dimension 1, 1, 2, 3, 3. Thus C can be realized as an S3-equivariantization of the 3-fermion theory.

Case 2:yz. Since all irreducibles must appear, we have that the stabilizers of y and z in S3 are trivial. Applying the orbit-stabilizer theorem, we can conclude that CS3 is a rank 13 modular category of dimension 1 + 6a2 + 6b2, where a = d3/6 and b = d4/6. In the case that CS3 is integral, by exhaustive search, we conclude that a = b = 1.11 On the other hand, if CS3 is not integral in which case we may apply Lemma III.8 to conclude that a = 1. Invoking the universal grading,21 we have b = 1 or b=7. In the latter case, CS3 has dimension 49, a contradiction24 (Proposition 8.32). Thus CS3 is pointed.

To see that this is not possible, let D be a rank 13 pointed modular category. Of course, the orthogonal group on Z13, with quadratic form q coming from the twist on D, OZ13,q, is either trivial or isomorphic to Z2. So the actions of S3 on D are given by elements of HomS3,OZ13,q, which are either trivial or the sign action. In the case of the trivial action, the isomorphism classes of simple objects in D are fixed under the action of S3. In particular, the de-equivariantization has rank 39. In the case of the sign action, there are 7 orbits of simples under the S3 action, and hence, the de-equivariantization has rank 21. Thus, C is not the S3-equivariantization of a rank 13 pointed modular category.

Proposition III.11.

IfCis a rank 5 properly pseudo-unitary premodular category with a rank 2 Müger center, then one of the following is true:

  • Cis Grothendieck equivalent to Rep(D8) withCRep(Z2). Furthermore, the T-matrix is of the formT=diag1,1,θ,1,1for some root of unity θ, corresponding to that 2-dimensional simple object, satisfying a monic quartic polynomial overZor

  • Cis Grothendieck equivalent toRepD14.

  • Cis Grothendieck equivalent toSU(2)8/Z2.

Moreover, each is realized (i) by aZ2-equivariantization of the toric code, (ii) by the adjoint subcategory of a 56-dimensional metaplectic category, e.g.,SO142ad, and (iii) bySU28/Z2.

Proof.

If a group G has only two irreducible representations, then GZ211 [Lemma 4.4(ii)]. Thus C is Grothendieck equivalent to RepZ2. Since C has odd rank, we can further conclude that C is Tannakian. Ordering the simples so that X0=I and X1 generates C, we have two cases to consider.

Case 1:X1X3 = X4 and X1X2 = X2. It follows immediately from these fusion rules that X2 is self-dual. Next note that the de-equivariantization CZ2 is a modular category of rank 4 and two of the objects have the same dimension. This forces CZ2 to be pointed. Since C is pseudo-unitary, we either have Fib ⊠ Fib, Fib ⊠ Sem, or FibSem¯.9 

In any of these cases, we get d2 = 2 and d3 = d4 = 1. Thus X2X2=IX1X1X2 or IX1X3X4. In the former case, X2 would generate a 6-dimensional fusion subcategory. This is not possible as 6 ∤ 8. Thus X2 generates C. Next note that this category is necessarily near group and hence can only be braided if X3*=X425,26 (Remark 4.4). It now follows that C is Grothendieck equivalent to RepD8.9 Examining the balancing equation for S1,3, we see that θ3 = θ4. Since X2 is self-dual, we can conclude that column 2 in S is real. In particular, θ3 = ±1. Since X3 is not in the Müger center of C, we know that the 3-rd and 0-th column of S must be orthogonal; hence, θ3 = θ4 = −1. From here, we can conclude that
θ2+θ2=1D2b,cNb,c2dbdcθbθc2Z  8(Corollary 3.3).
(3)

This yields a quartic polynomial for θ over Z. Considering all possible n such that ϕn<4 gives possible primitive roots. However, θ = ζn with ϕn<4 satisfies (3). Such a category can be constructed by considering the equivariantization of the toric code under the nontrivial Z2-action.

In the case where CZ2=FibFib, we get the following set of dimensions {1, 1, φ2, φ2, 2φ} for C, where φ is the golden mean. After computing the fusion rules using (1), we can conclude that C is Grothendieck equivalent SU(2)8.

If CZ2=FibSem, we get a category C with dimensions {1, 1, 1, 1, 2φ}. This category violates a condition given for near-group categories25 (Theorem 1.1); hence, it does not exist. The case of FibSem¯ follows similarly.

Case 2: X1Xa = Xa for 2 ≤ a ≤ 4. In this case, the de-equivariantization is rank 7 modular with simples of dimension 1,d1^,d1^,d2^,d2^,d3^,d3^. In particular, we know that CZ2pt is either RepZ3 or CZ2 (Lemma III.8).12 In the former case, applying the universal grading and the pigeon-hole principle gives d^22+d^32=3. From here, notice that the global dimension of this category is 9, and we can conclude that CZ2pt=CZ2RepZ324 (Proposition 8.32), a contradiction.

Thus CZ2 is pointed, and hence, d1 = d2 = d3 = 2. By the pigeon-hole principle and without loss of generality, we may assume X2 is self-dual. Just as in case 1, a dimension argument shows that X2 cannot be a sub-object of X2X2. So without loss of generality, we have X2X2=IX1X3. Of course, the category generated by X2 cannot exclude X4 since its dimension must divide 14. Thus C is cyclically generated by X2. Furthermore, since X2 is self-dual, its tensor square must be as well. In particular, X3 is self-dual. The pigeon-hole principle reveals that X4 must also be self-dual. Thus C is Grothendieck equivalent to RepD1426 (Theorem 4.2). One can arrive at the same conclusion by considering equivariantizations of the rank 4 pointed modular categories,9 as in the proof of Proposition III.10. Such categories may be realized as the adjoint categories of 56-dimensional metaplectic categories.27 

Remark III.12.

It is interesting to note that none of the known algebraic conditions8for premodular categories are sufficient to determine θ in Proposition III.11 beyond it being a root of a quartic monic polynomial overZ.

The authors would like to thank César Galindo for enlightening discussions. The research described in this paper was, in part, conducted under the Laboratory Directed Research and Development Program at PNNL, a multi-program national laboratory operated by Battelle for the U.S. Department of Energy, Release No. PNNL-SA-120942.

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