Definite orders with locally free cancellation

We enumerate all orders in definite quaternion algebras over number fields with the Hermite property; this includes all orders with the cancellation property for locally free modules.


Introduction
Motivation. Let R be the ring of integers of a number field F with class group Cl R, the group of isomorphism classes of locally principal R-modules under tensor product. In fact, every finitely generated, locally free R-module M is of the form M ≃ a 1 ⊕ · · · ⊕ a n where each a i is a locally principal R-module; moreover, we have a 1 ⊕ · · · ⊕ a n ≃ b 1 ⊕ · · · ⊕ b m if and only if m = n and [a 1 · · · a n ] = [b 1 · · · b m ] ∈ Cl R. In particular, every such M is of the form M ≃ R m ⊕ a and the Steinitz class [a] ∈ Cl R is well-defined on the R-module isomorphism class of M. Thus, if a, b are locally principal R-modules and m 0, then: Generalizations. We now pursue a noncommutative generalization: we seek to define a group operation on isomorphism classes of modules in a way analogous to the previous section. Let O be an R-order in a finite-dimensional semisimple F -algebra B. The (right) class set Cls O of O is the set of locally principal right fractional O-ideals I ⊆ B under the equivalence relation I ∼ J if there exists α ∈ B × such that I = αJ. Equivalently, Theorem 1.3. Up to ring isomorphism, there are exactly 303 definite Hermite quaternion orders and exactly 247 with locally free cancellation.
The orders in Theorem 1.3 are listed in appendix B, along with detailed information about them; a computer-loadable version is available [SV19]. These orders arise in quaternion algebras over exactly 36 number fields F up to isomorphism. If we fix the fields F arising this way, we can refine our count by looking at orders up to R-algebra isomorphism, where R is the ring of integers of F (thereby distinguishing Galois conjugates): counted this way, there are exactly 375 definite Hermite quaternion R-orders and exactly 316 with locally free cancellation.
Theorem 1.3 can be seen as the culminating resolution to a very general class number 1 (or unique factorization) problem for central simple algebras over number fields.
Previous results. Vignéras [Vig76] showed that there are only finitely many isomorphism classes of definite, hereditary quaternion orders with locally free cancellation. She provided a numerical criterion characterizing locally free cancellation, but in fact this was shown to characterize Hermite orders by Smertnig [Sme15]. By use of Odlyzko discriminant bounds, Vignéras found that Hermite orders are only possible over number fields of degree at most 33, and then she classified them over quadratic and cyclic cubic fields. More recently, Hallouin-Maire [HM06] classified definite hereditary Hermite orders (they refer to Eichler orders, but like Vignéras only consider those Eichler orders of squarefree reduced discriminant) by a rather involved analysis, finding that they arise only for fields of degree at most 6. Finally, Smertnig [Sme15] completed the classification of definite hereditary orders with locally free cancellation.
The Hermite property has appeared in work by other authors in various guises. Estes-Nipp [EN89] and Estes [Est91b] considered factorization properties in quaternion orders, among them one they call factorization induced by local factorization (FLF ). After observing that the principal genus of O (as defined by Estes) consists precisely of the stably free right O-ideals, the theorem of Estes [Est91b,Theorem 1] shows that O has FLF if and only if O is Hermite, a connection first noted by Nipp [Nip75]. Estes-Nipp classify all 40 definite Hermite quaternion Z-orders [EN89, Table I]. Unlike the results of Vignéras and Hallouin-Maire, their classification is not restricted to hereditary orders-but it is only carried out over R = Z.
Every quaternion order O with # Cls O = 1 trivially has locally free cancellation, and all orders with # Cls O 2 were enumerated by Kirschmer-Lorch [KL16b].
The global function field case was studied by Denert-Van Geel [DVG86,DVG88]. In this case, because of the absence of archimedean places, there are central simple algebras of arbitrary dimension that do not satisfy strong approximation. Denert-Van Geel show that if O is a definite Hermite order over a global ring in a function field F , then F = F q (t) must be a rational function field [DVG86, Theorem 2.1]. Moreover, they prove that-unlike the number field setting-there are infinitely many nonisomorphic quaternion algebras containing definite maximal orders with locally free cancellation [DVG86, Theorem 2.2]. (Some care must be taken in reading these papers, as they seem to incorrectly identify the Hermite property with locally free cancellation.) Discussion. Our proof of Theorem 1.3 follows the general approach of Vignéras and Hallouin-Maire, but it differs in two important respects.
(1) Our paper is essentially self-contained, and in particular we do not use the criterion of Vignéras (derived from a computation involving Tamagawa measures). Instead, we give a simple, direct argument (Theorem 5.11, Corollary 5.13) that the mass of the fibers in the stable class map is constant; this allows us to reinterpret the theory in a clarifying way (Proposition 4.3). (For sanity, we show that our criterion implies the criterion of Vignéras, see Theorem A.1.) Combined with the Eichler mass formula, this provides a stable mass formula for each of the fibers of the stable class map, from which we may proceed with analytic estimates.
(2) As much as possible, we use machine computation in lieu of delicate, case-by-case analysis by hand. This makes it easier for the reader to verify and to experiment with the result [SV19]. It also means that we can get away using slightly weaker, but easier to prove, bounds for the degree of F (Proposition 6.3). This approach is important for reproducibility and to avoid slips, given the complexity of the answer. Our calculations are performed in the computer algebra system Magma [BCP97]; the total running time is less than an hour on a standard CPU. We hope that our quaternionic algorithms will be of further use to others, beyond the classification in this paper: for example, we implement a systematic enumeration of suborders and superorders of quaternion orders, as well as the computation of the stable class group. We checked our output by restricting it and comparing to existing lists of orders (see Remark 6.14), and in every case we checked they agree.
The list of definite quaternion orders is quite remarkable! For some interesting examples, further discussion, and an application to factorization, see section 7.
Organization. Our paper is organized as follows. In section 2, we set up preliminaries on class groups and in section 3 the properties of locally free cancellation and Hermite. In section 4, we characterize these properties in terms of masses. In section 5, we establish a mass formula for the fibers of the stable class map. Next, in section 6, we present bounds for the search and our algorithm to find all orders with locally free cancellation. We then discuss examples and applications in section 7. Finally, in Appendix A, we compare our stable mass formula with the criterion of Vignéras, and then in Appendix B we present the tables describing the output in detail.
Acknowledgements. The authors would like to thank Emmanuel Hallouin, Christian Maire, Markus Kirschmer, Roger Wiegand, and the anonymous referee. Voight was supported by an NSF CAREER Award (DMS-1151047) and a Simons Collaboration Grant (550029). Smertnig was supported by the Austrian Science Fund (FWF) project J4079-N32. The research for this paper was conducted while Smertnig was visiting Dartmouth College; he would like to extend his thanks for their hospitality.

Preliminaries
As a general reference for quaternion algebras, we refer to the books of Vignéras [Vig80] and Voight [Voi19].
Notation. Throughout, let F be a totally real number field of degree n = [F : Q] and ring of integers R. Let F × >0 be the set of totally positive elements of F × (positive under all real embeddings of F ). For a (nonzero) prime p of R, let F p denote the completion of F at p with valuation ring R p ⊆ F p . We write F := ′ p F p for the finite adeles of F , the restricted direct product with respect to R p indexed over the primes of R; we similarly write R := p R p .
Let S be a finite (possibly empty) set of primes of R. For a subset X ⊆ F , denote by X S,1 := {α = (α p ) p ∈ X : α p = 1 for all p ∈ S}.
(2.1) Let Idl R be the group of fractional R-ideals a ⊆ F , and let Idl S R Idl R be the subgroup of those a ∈ Idl R for which a p = R p for all p ∈ S. Let Just as there are canonical group isomorphisms the 'global' definition given in Definition 2.4 can also be given equivalently adelically as in the following lemma. We only use this lemma to identify the stable class group with Cl G(O) R to compare with work of Fröhlich [Frö75] later on.
Lemma 2.8. The following statements hold.
Next, (b). The middle equality is due to the obvious inclusion R × S,1 ⊆ F × >0 nrd( O × ) ∩ F × S,1 . By weak approximation, the homomorphism Class set and mass. We now turn to the class set of our order.
∈ Q 0 (2.13) (well-defined independent of the choice of representatives). We will later (Theorem 6.1) recall an explicit expression for mass(Cls O), generalizing the Eichler mass formula.
Finally, we define the genus of O to be the set of R-orders in B locally isomorphic to O.

Locally free cancellation
In this section, we now relate the class set to naturally associated abelian groups. As the following lemma shows, locally free cancellation actually implies the apparently stronger property that arbitrary locally free, finitely generated modules may be cancelled from direct sums, thus justifying the name.
Lemma 3.6. An order O has locally free cancellation if and only if for all locally free, finitely generated O-module M, N, K, we have That is, the commutative monoid of isomorphism classes of locally free, finitely generated O-modules under direct sum is cancellative.
Proof. The implication (⇐) is clear. For (⇒), suppose that O has locally free cancellation and M, N, K are such that K ⊕ M ≃ K ⊕ N. Since K is locally free, the fact that Ext 1 commutes with localization in our setting shows that K is projective; see, for example,   Reduced norms and equivalences. Define We recall the following local description of the stable class group.
In particular, Theorem 3.11 yields a commutative diagram where the top map is the stable class map (3.7) and the bottom map is the natural projection.
Proof. Apply the reduced norm to the right-hand side of (3.13) (an isomorphism of groups onto its image) and then apply Lemma 2.8.
Since Cl G(O) R is the most accessible of these groups, in the remainder of the paper we will always work with it.
As a consequence of these isomorphisms, we can interpret the fibers of the reduced norm map nrd : Cls O → Cl G(O) R as parametrizing the isomorphism classes of locally principal right fractional O-ideals in a given stable isomorphism class.

Characterization
In this section, we characterize the locally free cancellation and Hermite properties in terms of masses and compare the two properties.
Masses. We begin with a quick lemma.
To  We now prove the promised characterizations.
Proposition 4.3. The following are equivalent.
(i) O has locally free cancellation.
(ii) The monoid of isomorphism classes of locally free, finitely generated right O-modules is cancellative.
.  (i) O is a Hermite ring.
(ii) Every stably free right O-ideal is free.
Moreover, these statements are equivalent to the corresponding statements for left O-modules.
Proof. As in Proposition 4.3.
Further remarks. We give a few remarks on extensions and other characterizations of the above. Remark 4.7. Since O has Krull dimension 1, its stable rank (and hence its general linear rank), is at most 2 [MR01, 11.3.7]. From this it follows-whether or not O is Hermite-that every stably free module of rank 2 is free, a fact which can also be seen by an application of strong approximation. There is a stronger connection between the two properties than it might initially seem, as the following result shows.
Thus, IJ −1 is stably free, and by assumption on O ′ , IJ −1 = αO ′ is free with α ∈ B × . Multiplying on the right by J, we see I = αJ so I ≃ J as right O-ideals.

Masses and suborders
Our goal in this section is to investigate the behavior of mass(Cls  The result follows. For part (c), we apply (a) to get With the basic comparison of masses in hand, we prove the following key theorem.
Proof. We begin with (a). Let Proof. This is a consequence of the snake lemma applied to By Hensel's Lemma, 1 + p l R p ⊆ R ×2 p for l > v p (4). Therefore, 1 + p l R p ⊆ nrd(O × p ). We make use of this in applying weak approximation in the following two proofs.
Proposition 5.16. There is an exact sequence Observing that the following is independent of the representatives b, c, we define a → (sgn a, bc −1 ) (5.17) By weak approximation, f is surjective. Clearly ker f = F × S,O . Proposition 5.18. Let O ⊆ O ′ be orders. Then there is an exact sequence Proof. The first equality follows from Proposition 5.16. For the second, note that Proof. Noting that nrd(O ′× p ) = R × p , the result follows from Proposition 5.18.

Bounds and algorithmic considerations
In this section, we bound the set of definite orders with the Hermite property by an estimate of mass and Odlyzko bounds.
We have the following generalization of Eichler's mass formula to arbitrary definite orders.
Theorem 6.1 (Mass formula). We have Moreover, if n = 9, then d 1/9 F < 13.53. Proof. We repeat a simple variation of the proof of Hallouin-Maire for the convenience of the reader.
If O is a Hermite order, then the same is true for every order containing O, by Corollary 5.6. Thus, we may without restriction assume that O is maximal. Then nrd(O × p ) = R × p for all primes p of R, and hence Cl G(O) R = Cl + R is the narrow class group.
In Corollary 5.13, we showed that O is Hermite then Applying the mass formula (Theorem 6.1) and using the trivial estimates ζ F (2) 1 and Nm(N) p|N λ(O|p) 1, the first bound for the root discriminant (6.4) is obtained.
By Proposition 5.19, Substituting this into the first bound, we get the bound (6.5). Comparing (6.5) with the discriminant bounds for totally real fields found in Odlyzko's tables [Odl76], we conclude n 14.
The Hilbert class field of F is a totally real field of absolute degree nh(R) and with the same root discriminant as F . If Cl R is nontrivial, then the degree of the Hilbert class field is at least 2n. Again comparing with the Odlyzko bounds for totally real fields, we conclude Cl R must be trivial if n 8. The quotient Cl + R/ Cl R is an elementary abelian 2-group and the Armitage-Fröhlich theorem [AF67, I] states that Hence, in case n 8, we have #Cl + R/#Cl R = #Cl + R 2 ⌊n/2⌋ . Substituting into (6.4), we must have d 1/n F 2 4/3+2(⌊n/2⌋−1)/(3n) π 4/3 if n 8. Comparing this improved bound with the Odlyzko tables, we find n 10. Moreover, d 1/10 F 13.95 for n = 10 and d 1/9 F < 13.53 for n = 9. By Voight's tables of totally real fields [Voi08b], there are no such fields of degree 10. Hence, n 9.
Remark 6.7. Without Odlyzko's tables, Vignéras [Vig76] showed [F : Q] 33. By first bounding n 14 as above and then using the bound in (6.4) for each such n and different possibilities of #Cl + R/#Cl R, Hallouin-Maire show n 8 (without using Armitage-Fröhlich). Using more careful arguments, Hallouin-Maire actually show n 6 in [HM06, Proposition 11] and for these degrees obtain improved bounds on the discriminant in [HM06,Proposition 12]. Since we leave the ultimate classification up to a computer, a priori we only need bounds that are good enough to ensure that all fields in questions have been tabulated. The weaker bounds that are more easily obtained suffice for us. .
The previous bounds imply the following finiteness result, proven by Vignéras [Vig76].
Proof. Since F is a totally real field with bounded root discriminant by Proposition 6.3, there exist only finitely many such fields. For each field, there exist only a finite number of possible choices for the ramified places of B by Proposition 6.8. Finally, for each of the finitely many isomorphism classes of maximal orders in B, the index of a Hermite suborder is bounded by Lemma 6.9.
Algorithm. We are now in a position to state an algorithm that finds all definite Hermite orders.
Algorithm 6.11. The following algorithm enumerates all definite Hermite orders.
1. Using the tabulation of totally real fields [Voi08a,Voi08b], enumerate all fields with n = [F : Q] 9 and, d 1/n F < 16.4 if n 8, respectively d 1/n F < 13.53 if n = 9. For each eligible field F , use Proposition 6.8 to compute a finite list of primes p at which B can ramify. 2. For each such algebra B, determine a set of representatives for the isomorphism classes of maximal orders. 3. For each Hermite maximal order O ′ , using Lemma 6.9, compute a list of prime ideals of R at which we need to consider non-maximal orders. 4. Iteratively compute suborders of O ′ at the given primes, using Lemma 6.9 to bound the necessary index, and check them for the desired property by computing their stable class group.

Proof of correctness. The bound in
Step 1 is valid by Proposition 6.3. For Step 2, we refer to Kirschmer-Voight [KV10,KV12]. In Step 4, to check whether a given order is a Hermite ring, we compute the stable class group Cl G(O) R and mass(Cls O) and use Proposition 4.4(iv).
The enumeration of suborders and the computation of Cl G(O) R are not readily available in existing computer algebra systems. Thus, we give some more detail on how these steps can be implemented efficiently.
Remark 6.12 (Computation of Cl G(O) R). Since algorithms to compute ray class groups are already implemented in computer algebra systems, it is easiest to compute Cl G(O) R as a quotient of such a group.
1. We compute the stable class group as a quotient of a ray class group. For a prime ideal p of R, define l(p) = v p (4) + 1. Let m = p∈S p l(p) , and let F × m = {a ∈ F × >0 : a ≡ 1 mod m}. By choice of l(p) and Hensel's Lemma, 1 + p l(p) R p ⊆ R ×2 p ⊆ nrd(O × p ). Thus, F × m ⊆ F × S,O and the stable class group can be realized as a quotient of the ray class group Cl + m R = Idl S R/F × m . More precisely, there is an exact sequence As in Proposition 5.18, To compute Cl G(O) R, we therefore first compute Cl + m R, and then compute, for each p ∈ S, a set of generators for nrd(O × p )/(1 + p l(p) R p ). Using the Chinese Remainder Theorem to obtain suitable global representatives for these generators, we compute Cl G(O) R as quotient of Cl + m R. 2. To compute nrd(O × p )/(1 + p l(p) R p ), first note that nrd(1 + p l(p) O p ) ⊆ 1 + p l(p) R p . Hence, the reduced norm induces a homomorphism Thus, it suffices to compute the image of a generating set of O × p /(1 + p l(p) O p ) under nrd.
Since O × p ⊇ 1 + pO p 1 + p 2 O p · · · 1 + p l(p) O p , it suffices to compute generating sets of the multiplicative groups O × p /(1 + pO p ) ≃ (O p /pO p ) × ≃ (O/pO) × and (1 + p i O p )/(1 + p i+1 O p ). The latter group is isomorphic to the additive group O/pO. A generating set for (O/pO) × can be computed since O/pO is a finite-dimensional algebra over the finite field R/p. A Z-basis of O yields a generating set for the additive abelian group O/pO. Remark 6.13 (Enumeration of suborders). To enumerate suborders in a systematic way, we organize them by radical idealizers [Voi19, Section 24.4]. We proceed as follows.
1. Compute representatives for the isomorphism classes of all maximal orders, giving a list of orders. 2. For each prime p for which we need to consider non-maximal orders, and for each (p-maximal) order O computed so far: a. First compute the hereditary suborders that are non-maximal at p. b. Recursively compute all suborders whose radical idealizer at p is one of the orders computed so far. For a given prime p, this procedure produces a tree of orders. If an order O exceeds the index bound or fails to be Hermite, we need not check its suborders anymore due to Corollary 5.6.
If O ′ is an order, and O ⊆ O ′ is a suborder whose radical idealizer at p is O ′ , then pO ′ ⊆ O. Hence, to compute candidate orders O, we simply check the preimages of all the subrings of the (finite) R/p-algebra O ′ /pO ′ . We use isomorphism testing between those orders to enumerate the orders up to isomorphism.
We have implemented this algorithm in Magma [BCP97]. Our code is available on the web [SV19]. Running this classification gives all definite quaternion orders over a ring of algebraic integers that are Hermite rings. From this list it is easy to filter the ones having locally free cancellation.
Remark 6.14. We checked our results against existing lists.
• The hereditary definite quaternion orders that are Hermite rings have been classified by Vignéras [Vig76], Hallouin-Maire [HM06], and Smertnig [Sme15]. Up to R-algebra isomorphism there are 168 such orders, of which 149 have locally free cancellation. Our list is consistent with the (corrected) old classification. • Kirschmer-Lorch [KL16b] computed all definite quaternion orders with type number at most 2, and made them available in electronic form [KL16a]. This includes all orders O with #Cls O = 1, and such an order trivially has locally free cancellation. Our list is consistent with theirs. • Estes-Nipp [EN89, Table I] list the 40 definite Hermite quaternion Z-orders. We found the same number of Z-orders, and we matched their discriminants to the ones appearing in our list.
Corollary 6.15. Up to R-algebra isomorphism, there are 375 definite Hermite quaternion orders; of these, 316 have locally free cancellation.
Reducing this list to orders up to ring isomorphism by Galois automorphisms, we obtain Theorem 1.3. Invariants describing these orders are given in Appendix B. A computerreadable file containing all the orders is available on the web [SV19].
Looking over the list in Appendix B, we find the following corollary.
Corollary 6.16. If a definite quaternion order O has locally free cancellation, then the base ring R is factorial.
Note that the analogous statement fails to hold for Hermite orders, with the sole exceptions being two orders over R = Z[ √ 15].

Examples and applications
Examples. Example 7.1. First, let B be the usual Hamiltonian quaternion algebra with i 2 = j 2 = −1 and ij = −ji. Then O = Z + 2Zi + 2Zj + 2Zij is an order of reduced discriminant 32 that has locally free cancellation and is non-Gorenstein. It has #Cls O = #StCl O = 2 but type number 1. Under the bijection between isomorphism classes of quaternion orders and similarity classes of ternary quadratic forms, O corresponds to 2(x 2 + y 2 + z 2 ). Its Gorenstein saturation, corresponding to x 2 + y 2 + z 2 , is the Lipschitz order Z + Zi + Zj + Zij.
There is also an interesting connection with quaternary quadratic forms, discussed in more detail in the next subsection. We restrict to the case R = Z for illustration. Let O be a definite quaternion Z-order.  [Par74] classified all 40 definite quaternion Z-orders with spinor class number 1, extending the list of 39 orders with # Cls(nrd| O ) = 1 (that is, class number 1) previously determined by Pall [Pal46], with the sole outlier described as follows. The order O corresponds to the ternary quadratic form x 2 + 3y 2 + 3z 2 − 3yz; On the other hand, the quaternary quadratic form nrd| O has discriminant 729 = 27 2 and in the basis above is given by Q(x, y, z, w) = x 2 + 7y 2 + 9z 2 + 3w 2 + xy + 9zw. (7.3) We have SpCls(Q) = 1 but Cls(Q) = 1, indeed Cls(Q) = 3 and Gen(Q) splits into two spinor genera. In fact, the form Q represents the unique class of primitive quaternary quadratic form with # SpCls(Q) = 1 but # Cls(Q) = 1 [EH19].
More generally, the classification of definite quadratic integral lattices with class number 1, started by Watson in the 1960s, has recently been finished by Lorch-Kirschmer [LK13]. (The rank 2 case assumes GRH.) Earnest-Haensch [EH19] conclude that the lattice found by Parks remains the sole example with spinor class number 1 but not class number 1 (by completing the classification for rank 4).
An application to factorizations. Aside from the intrinsic importance of the Hermite and cancellation properties for the description of isomorphism classes of locally free modules, the Hermite property has been shown to have important consequences for the factorizations of elements in an order. This has been observed by Estes-Nipp in [EN89,Est91b] in their study of factorizations induced by norm factorization (FNF ), as well as more recently in studying non-unique factorizations in orders by means of arithmetical invariants.
We give a brief glimpse at this connection, by highlighting some properties of the sets of lengths. Let O • be the multiplicative monoid of non-zero-divisors of O, that is, the elements of non-zero reduced norm. An element u ∈ O • is an atom if it cannot be expressed as a product of two non-units. Every non-unit a ∈ O • can be written as a product of atoms, but in general not uniquely so. If a = u 1 · · · u k , with atoms u i , then k is a length of a, and we write L(a) ⊆ Z 0 for the set of lengths of a. By considering the reduced norm, it is easily seen that these sets are finite. If #L(a) 2, then #L(a k ) k + 1. Hence, the sets of lengths are either all singletons, in which case O • is half-factorial, or become arbitrarily large. If holds. This is very close to Vignéras's criterion [Vig76] but not precisely the same. In this appendix, we show how to derive her original criterion from ours. (1 + Nm p) 22 We will need the following small lemma.
We first consider the case where O is hereditary. From the mass formula (Theorem 6.1), and Proposition 5.19, .
Here, the sign can be expressed as (−1) n , because B is ramified at an even number of places, since B is definite it is ramified at all archimedean places, and D is the product of all nonarchimedean places at which B is ramified. Moreover, we have Putting everything together, the claim follows.

24
Appendix B. Tables The following tables list invariants describing all definite quaternion orders O that are Hermite rings, hence all orders with locally free cancellation. For simplicity, orders are listed up to ring isomorphism not up to R-algebra isomorphism. That is, we use automorphisms of the base field to identify some of the orders. The corresponding multiplicity is listed in the last column of the tables. Thus, there are 303 entries in the tables, but 375 orders up to R-algebra automorphism.
For example, in R = Z[(1 + √ 5)/2] the prime number 11 splits, so that 11R = pq. Let B be the quaternion algebra over F = Q[ √ 5] that is ramified only at the archimedean primes (corresponding to n = 2, d = 5, D = 1 in the table below). In B there exist two hereditary orders O, O ′ with N := Nm(discrd(O)) = Nm(discrd(O ′ )) = 11 having cancellation. One of these is maximal at p and non-maximal at q, while the other one is non-maximal at p and maximal at q. However, the Galois automorphism of F maps p to q, and, extending it to a ring automorphism of B, it maps O to O ′ . Thus, O and O ′ are isomorphic as rings but not as R-algebras. We record only one entry in the table (the line with n = 2, d = 5, D = 1, N = 11), but note the multiplicity 2 in the column labeled by '#'.
In the tables, we use the following notation.
• n = [F : Q] is the degree of the base field F . An empty entry in one of these columns means that the corresponding structure (quaternion algebra, respectively, the base field), is the same as in the previous line. For the number fields appearing in the tables, the pair (n, d) uniquely characterizes them up to field isomorphism.
For a prime p of R, let k := R/p be the residue class field of p. Recall that the Eichler symbol of a quaternion R-order O is defined as if O p / rad O p ≃ k; and −1, if O p / rad O p is a (separable) quadratic field extension of k.
The next column lists the strongest property that the order possesses (globally), among the following: hereditary Eichler maximal residually quadratic Bass Gorenstein.

residually inert
The column labeled 'c' contains an entry 'c' whenever the order listed has locally free cancellation, and is empty otherwise (that is, the order is a Hermite order but does not have locally free cancellation).
Next we list local Eichler symbols. The data are organized by rational primes. For each prime p, if there is a prime ideal p of R over p for which O p is not maximal, we list the local Eichler symbol (O | p) for all p over p. Here, the primes lying over p are sorted first in ascending order of Nm p, then by ascending ramification index. For easier readability, if O is maximal at every p containing p, we do not list any data. This means (O | p) = * if the quaternion algebra is unramified at p, and (O | p) = −1 if the algebra is ramified at p.
Next, we list the cardinality of the class set of locally free right ideals of O, of the stable class group of O (which is isomorphic to Cl G(O) R), and the class group of R itself. If the value is 1, we omit the entry for easier readability. Next, we list the cardinality of the class set of locally free right ideals of O, of the stable class group of O (which is isomorphic to Cl G(O) R), the type number t(O), and the class group of R itself. If the value is 1, we omit the entry for easier readability.
The type number is the number of isomorphism classes of orders that are locally isomorphic to O; again we suppress the value 1. Observe that if t(O) > 1 and O has cancellation, then the orders that are locally isomorphic but not isomorphic to O also have cancellation, and therefore also appear in the tables. Several instances of this can be seen over in the algebra with N = 1 over the field Q[ √ 12]. The final column lists the multiplicity with which the entry should be counted, to count the orders up to R-algebra isomorphism. Again, if the value is 1, we omit the entry.
The tables are intended to give an overview over the orders, not to characterize them up to isomorphism. A computer-readable list of all the orders in the table, including generators, is available electronically at [SV19].