..nodoctest
Bisson-Streng¶
x.__init__(...) initializes x; see help(type(x)) for signature
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recip.bissonstreng.CM_type_to_mus(Phi, F)¶ Returns mu generating N_Phir(aaa) with mu*mubar in QQ for aaa ranging over generators of the ray class group mod F of the reflex field of Phi.
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recip.bissonstreng.all_period_matrices_two(lst)¶ Find all period matrices with CM by an order of 2-power index in OK such that the field of moduli is in Kr, where K ranges over the fields given by lst.
And tests whether for all but [5,5,5] that there are no orders with non-2-power index such that the field of moduli is in Kr (by raising an error if there is such an order).
OUTPUT:
- List of triples (K, Phi, F, Zs, orders), where Zs[k] is a period matrix has CM by orders[k] for each k, and F*OK is contained in each element of orders.
EXAMPLES:
sage: from recip import * sage: lst = wamelen_dab_list() + [[5, 15, 45], [5, 30, 180], [5, 35, 245], [5, 105, 2205], [8, 12, 18], [17, 119, 3332], [17, 255, 15300]] sage: Zs = all_period_matrices_two(lst) # long time, 2 minutes sage: [len(l) for (K,Phi,F,l,orders) in Zs] # long time [6, 3, 1, 6, 2, 1, 2, 1, 6, 2, 6, 1, 1, 4, 12, 4, 4, 6, 2, 4] sage: matr = [two_two_matrix(l, F) for (K, Phi, F, l, orders) in Zs] # long time sage: matr # long time, output is random [ [0 1 1 1 0 0] [0 1 1 0 1 0] [1 0 0 0 1 1] [1 0 0 1 0 1] [1 0 0 0 1 1] [1 0 0 1 0 1] [1 0 0 0 1 1] [1 1 1] [0 1 1 0 1 0] [0 1 1 1 0 0] [1 1 1] [1 0 0 1 0 1] [0 0] [0 0] [0 1 1 1 0 0], [1 1 1], [0], [0 1 1 0 1 0], [0 0], [0], [0 0], [0], [1 0 0 1 0 1] [0 1 1 0 1 0] [0 1 1 0 1 0] [1 0 0 1 0 1] [0 1 1 0 1 0] [1 0 0 1 0 1] [0 1 0 0] [1 0 0 1 0 1] [0 1 1 0 1 0] [1 0 0 0] [0 1 1 0 1 0] [0 0] [1 0 0 1 0 1] [0 0 0 1] [1 0 0 1 0 1], [0 0], [0 1 1 0 1 0], [0], [0], [0 0 1 0], [0 0 1 0 1 0 1 0 0 0 0 0] [0 0 0 1 0 1 0 1 0 0 0 0] [1 0 0 0 0 0 0 0 1 0 1 0] [0 1 0 0 0 0 0 0 0 1 0 1] [1 0 0 0 0 0 0 0 1 0 1 0] [0 1 0 0 0 0 0 0 0 1 0 1] [1 0 0 0 0 0 0 0 1 0 1 0] [1 0 0 1 0 1] [0 1 0 0 0 0 0 0 0 1 0 1] [0 1 1 0 1 0] [0 0 1 0 1 0 1 0 0 0 0 0] [0 0 0 1] [0 0 0 0] [0 1 1 0 1 0] [0 0 0 1 0 1 0 1 0 0 0 0] [0 0 1 0] [0 0 0 0] [1 0 0 1 0 1] [0 0 1 0 1 0 1 0 0 0 0 0] [0 1 0 0] [0 0 0 0] [0 1 1 0 1 0] [0 0] [0 0 0 1 0 1 0 1 0 0 0 0], [1 0 0 0], [0 0 0 0], [1 0 0 1 0 1], [0 0], [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] ]
The output above was with an older version of Sage. We now check that the actual output agrees up to graph isomorphism, and then we apply a graph isomorphism and continue with the result after isomorphism.:
sage: L = [[[0, 1, 1, 1, 0, 0], ....: [1, 0, 0, 0, 1, 1], ....: [1, 0, 0, 0, 1, 1], ....: [1, 0, 0, 0, 1, 1], ....: [0, 1, 1, 1, 0, 0], ....: [0, 1, 1, 1, 0, 0]], ....: [[1, 1, 1], [1, 1, 1], [1, 1, 1]], ....: [[0]], ....: [[0, 1, 1, 0, 1, 0], ....: [1, 0, 0, 1, 0, 1], ....: [1, 0, 0, 1, 0, 1], ....: [0, 1, 1, 0, 1, 0], ....: [1, 0, 0, 1, 0, 1], ....: [0, 1, 1, 0, 1, 0]], ....: [[0, 0], [0, 0]], ....: [[0]], ....: [[0, 0], [0, 0]], ....: [[0]], ....: [[1, 0, 0, 1, 0, 1], ....: [0, 1, 1, 0, 1, 0], ....: [0, 1, 1, 0, 1, 0], ....: [1, 0, 0, 1, 0, 1], ....: [0, 1, 1, 0, 1, 0], ....: [1, 0, 0, 1, 0, 1]], ....: [[0, 0], [0, 0]], ....: [[0, 1, 1, 1, 0, 0], ....: [1, 0, 0, 0, 1, 1], ....: [1, 0, 0, 0, 1, 1], ....: [1, 0, 0, 0, 1, 1], ....: [0, 1, 1, 1, 0, 0], ....: [0, 1, 1, 1, 0, 0]], ....: [[0]], ....: [[0]], ....: [[0, 1, 0, 0], [1, 0, 0, 0], [0, 0, 0, 1], [0, 0, 1, 0]], ....: [[0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0], ....: [0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0], ....: [1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0], ....: [0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1], ....: [1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0], ....: [0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1], ....: [1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0], ....: [0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1], ....: [0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0], ....: [0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0], ....: [0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0], ....: [0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0]], ....: [[0, 0, 0, 1], [0, 0, 1, 0], [0, 1, 0, 0], [1, 0, 0, 0]], ....: [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]], ....: [[1, 0, 0, 1, 0, 1], ....: [0, 1, 1, 0, 1, 0], ....: [0, 1, 1, 0, 1, 0], ....: [1, 0, 0, 1, 0, 1], ....: [0, 1, 1, 0, 1, 0], ....: [1, 0, 0, 1, 0, 1]], ....: [[0, 0], [0, 0]], ....: [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]]] sage: matrL = [Matrix(c) for c in L] sage: matr == matrL # exact equality, was False at latest Sage version, long time False sage: t = [(Graph(matrL[c]).is_isomorphic(Graph(matr[c]), certificate=True), matr[c]) for c in range(len(L))] # long time sage: all([a[0] for (a,b) in t]) # here is what matters, long time True sage: matr = [Matrix([[c[a[1][i],a[1][j]] for j in range(c.ncols())] for i in range(c.nrows())]) for (a,c) in t] # long time sage: matr == matrL # now we do have the same graph as before (when forgetting the orders, so still doctests can fail at later stages in later versions of Sage). long time True sage: matr # long time [ [0 1 1 1 0 0] [0 1 1 0 1 0] [1 0 0 0 1 1] [1 0 0 1 0 1] [1 0 0 0 1 1] [1 0 0 1 0 1] [1 0 0 0 1 1] [1 1 1] [0 1 1 0 1 0] [0 1 1 1 0 0] [1 1 1] [1 0 0 1 0 1] [0 0] [0 0] [0 1 1 1 0 0], [1 1 1], [0], [0 1 1 0 1 0], [0 0], [0], [0 0], [0], [1 0 0 1 0 1] [0 1 1 1 0 0] [0 1 1 0 1 0] [1 0 0 0 1 1] [0 1 1 0 1 0] [1 0 0 0 1 1] [0 1 0 0] [1 0 0 1 0 1] [1 0 0 0 1 1] [1 0 0 0] [0 1 1 0 1 0] [0 0] [0 1 1 1 0 0] [0 0 0 1] [1 0 0 1 0 1], [0 0], [0 1 1 1 0 0], [0], [0], [0 0 1 0], [0 0 1 0 1 0 1 0 0 0 0 0] [0 0 0 1 0 1 0 1 0 0 0 0] [1 0 0 0 0 0 0 0 1 0 1 0] [0 1 0 0 0 0 0 0 0 1 0 1] [1 0 0 0 0 0 0 0 1 0 1 0] [0 1 0 0 0 0 0 0 0 1 0 1] [1 0 0 0 0 0 0 0 1 0 1 0] [1 0 0 1 0 1] [0 1 0 0 0 0 0 0 0 1 0 1] [0 1 1 0 1 0] [0 0 1 0 1 0 1 0 0 0 0 0] [0 0 0 1] [0 0 0 0] [0 1 1 0 1 0] [0 0 0 1 0 1 0 1 0 0 0 0] [0 0 1 0] [0 0 0 0] [1 0 0 1 0 1] [0 0 1 0 1 0 1 0 0 0 0 0] [0 1 0 0] [0 0 0 0] [0 1 1 0 1 0] [0 0] [0 0 0 1 0 1 0 1 0 0 0 0], [1 0 0 0], [0 0 0 0], [1 0 0 1 0 1], [0 0], [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] ]
The matrices above correspond to the (2,2)-isogeny graphs, one matrix for every order, one row/column for every period matrix. Next, we look at the orders corresponding to these period matrices. An entry a in row i and column j means that a is non-zero if and only if period matrices i and j have the same endomorphism ring, and a is the index of that endomorphism ring in the maximal order.:
sage: vis = [visualize_orders(orders) for (K, Phi, F, l, orders) in Zs] # long time sage: flatten([[Matrix(lst[k]),Matrix([[vis[k][t[k][0][1][i],t[k][0][1][j]] for j in range(vis[k].ncols())] for i in range(vis[k].nrows())]), matr[k]] for k in range(len(matr))]) # long time [ [ 1 0 0 0 0 0] [0 1 1 1 0 0] [ 0 4 0 0 0 0] [1 0 0 0 1 1] [ 0 0 8 8 0 0] [1 0 0 0 1 1] [ 0 0 8 8 0 0] [1 0 0 0 1 1] [1 0 0] [1 1 1] [ 0 0 0 0 16 16] [0 1 1 1 0 0] [0 2 2] [1 1 1] [5 5 5], [ 0 0 0 0 16 16], [0 1 1 1 0 0], [8 4 2], [0 2 2], [1 1 1], [1 1 0 0 0 0] [0 1 1 0 1 0] [1 1 0 0 0 0] [1 0 0 1 0 1] [0 0 4 4 4 4] [1 0 0 1 0 1] [0 0 4 4 4 4] [0 1 1 0 1 0] [0 0 4 4 4 4] [1 0 0 1 0 1] [13 13 13], [1], [0], [ 5 10 20], [0 0 4 4 4 4], [0 1 1 0 1 0], [1 1] [0 0] [ 5 65 845], [1 1], [0 0], [29 29 29], [1], [0], [ 5 85 1445], [1 1 0 0 0 0] [1 1 0 0 0 0] [0 0 2 2 2 2] [0 0 2 2 2 2] [1 1] [0 0] [0 0 2 2 2 2] [1 1], [0 0], [ 37 37 333], [1], [0], [ 8 20 50], [0 0 2 2 2 2], [1 0 0 1 0 1] [1 1 0 0 0 0] [0 1 1 0 1 0] [1 1 0 0 0 0] [0 1 1 0 1 0] [0 0 4 4 4 4] [1 0 0 1 0 1] [0 0 4 4 4 4] [0 1 1 0 1 0] [1 1] [0 0] [0 0 4 4 4 4] [1 0 0 1 0 1], [ 13 65 325], [1 1], [0 0], [13 26 52], [0 0 4 4 4 4], [0 1 1 1 0 0] [1 0 0 0 1 1] [1 0 0 0 1 1] [1 0 0 0 1 1] [0 1 1 1 0 0] [0 1 1 1 0 0], [53 53 53], [1], [0], [ 61 61 549], [1], [0], [1 1 1 1] [0 1 0 0] [1 1 1 1] [1 0 0 0] [1 1 1 1] [0 0 0 1] [ 5 15 45], [1 1 1 1], [0 0 1 0], [ 5 30 180], [1 1 1 1 0 0 0 0 0 0 0 0] [0 0 1 0 1 0 1 0 0 0 0 0] [1 1 1 1 0 0 0 0 0 0 0 0] [0 0 0 1 0 1 0 1 0 0 0 0] [1 1 1 1 0 0 0 0 0 0 0 0] [1 0 0 0 0 0 0 0 1 0 1 0] [1 1 1 1 0 0 0 0 0 0 0 0] [0 1 0 0 0 0 0 0 0 1 0 1] [0 0 0 0 4 4 4 4 4 4 4 4] [1 0 0 0 0 0 0 0 1 0 1 0] [0 0 0 0 4 4 4 4 4 4 4 4] [0 1 0 0 0 0 0 0 0 1 0 1] [0 0 0 0 4 4 4 4 4 4 4 4] [1 0 0 0 0 0 0 0 1 0 1 0] [0 0 0 0 4 4 4 4 4 4 4 4] [0 1 0 0 0 0 0 0 0 1 0 1] [0 0 0 0 4 4 4 4 4 4 4 4] [0 0 1 0 1 0 1 0 0 0 0 0] [0 0 0 0 4 4 4 4 4 4 4 4] [0 0 0 1 0 1 0 1 0 0 0 0] [0 0 0 0 4 4 4 4 4 4 4 4] [0 0 1 0 1 0 1 0 0 0 0 0] [0 0 0 0 4 4 4 4 4 4 4 4], [0 0 0 1 0 1 0 1 0 0 0 0], [ 5 35 245], [1 1 1 1] [0 0 0 1] [1 1 1 1] [0 0 0 0] [1 1 1 1] [0 0 1 0] [1 1 1 1] [0 0 0 0] [1 1 1 1] [0 1 0 0] [1 1 1 1] [0 0 0 0] [1 1 1 1], [1 0 0 0], [ 5 105 2205], [1 1 1 1], [0 0 0 0], [1 1 0 0 0 0] [1 0 0 1 0 1] [1 1 0 0 0 0] [0 1 1 0 1 0] [0 0 2 2 2 2] [0 1 1 0 1 0] [0 0 2 2 2 2] [1 0 0 1 0 1] [0 0 2 2 2 2] [0 1 1 0 1 0] [1 1] [ 8 12 18], [0 0 2 2 2 2], [1 0 0 1 0 1], [ 17 119 3332], [1 1], [1 1 1 1] [0 0 0 0] [1 1 1 1] [0 0 0 0] [0 0] [1 1 1 1] [0 0 0 0] [0 0], [ 17 255 15300], [1 1 1 1], [0 0 0 0] ]
The first matrix is for \(\QQ[\zeta_5]\), and from the others, we conclude Lemma 6.6. For \(\QQ[\zeta_5]\), we see that the graph is connected.
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recip.bissonstreng.is_S_O_equal_S_OK(O)¶ Given an order O in a non-biquadratic quartic CM-field O, returns True if and only if S_O = S_{O_K}, in the notation of Bisson-Streng.
EXAMPLES:
The same example as in is_trivial_in_shimura_group, but much faster:
sage: from recip import * sage: K = CM_Field([149,13,5]) sage: P.<x> = QQ[] sage: [alpha1,alpha2]=(x^4+2*x^3+16*x^2+15*x+19).roots(K, multiplicities=False) sage: O = K.order(alpha1) sage: is_S_O_equal_S_OK(O) False
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recip.bissonstreng.minimal_order_cl_nr_one_F(Phi, F=None, output_type='order', Ostart=None)¶ Returns an order O in the domain K of Phi such that for any order O’ in K containing ZZ+F*OK, if the image of I_Kr(F) in CCC(O) is trivial, then O contains O’.
Returns None every O’ satisfies this property, i.e., if O does not exist.
If K is not QQ(zeta_5), then actually the “if” in the first paragraph is an “if and only if”.
Here F is an integer.
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recip.bissonstreng.minimal_orders_of_class_number_one(Phi, output_type='order')¶ Let K be the domain of the input CM-type Phi. If K is not \(\QQ(\zeta_5)\), returns the list of tuples (p, k, i, O), where p ranges over all primes such that there is an order that is not p-maximal, but does have Shimura class number one. Here O is the smallest such order of index a power i of p, and \(p^k\) is the exponent of the additive group OK/O.
If K is \(\QQ(\zeta_5)\), then the output is as above, except that the list may contain too many primes p, and the orders O may be too small (in other words, for a classification, the output will be on the safe side).
EXAMPLES:
sage: from recip import * sage: lst = wamelen_dab_list() sage: K = CM_Field([5,5,5]) sage: Phi = K.CM_types()[0] sage: minorders = minimal_orders_of_class_number_one(Phi); minorders # long time 1 second [(2, 2, 16, Order in CM Number Field in alpha with defining polynomial x^4 + 5*x^2 + 5), (3, 1, 9, Order in CM Number Field in alpha with defining polynomial x^4 + 5*x^2 + 5), (5, 3, 15625, Order in CM Number Field in alpha with defining polynomial x^4 + 5*x^2 + 5)] sage: Phi = CM_Field(lst[1]).CM_types()[0] sage: m = minimal_orders_of_class_number_one(Phi); m [(2, 1, 4, Order in CM Number Field in alpha with defining polynomial x^4 + 4*x^2 + 2)]
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recip.bissonstreng.orders_proven_with_wamelen(latex_output=False)¶ Compute the orders O’‘’ in the article, and prove that the non-maximal O’‘’ have index 2 in the maximal order and no proper polarized ideal classes. Then output Table 1 of the article.
EXAMPLES:
sage: from recip import * sage: orders_proven_with_wamelen() [([5, 5, 5], 1, 1, 1, 1, 1), ([8, 4, 2], 1, x^4 + 4*x^2 + 2, 1, 1, 1), ([13, 13, 13], 1, x^4 - x^3 + 2*x^2 + 4*x + 3, 3, 1, 1), ([5, 10, 20], 2, x^4 + 10*x^2 + 20, 4, 4, 2), ([5, 65, 845], 2, x^4 - x^3 + 16*x^2 - 16*x + 61, 19, 1, 1), ([29, 29, 29], 1, x^4 - x^3 + 4*x^2 - 20*x + 23, 7, 1, 1), ([5, 85, 1445], 2, x^4 - x^3 + 21*x^2 - 21*x + 101, 29, 1, 1), ([37, 37, 333], 1, x^4 - x^3 + 5*x^2 - 7*x + 49, 21, 3, 1), ([8, 20, 50], 2, x^4 + 20*x^2 + 50, 25, 25, 1), ([13, 65, 325], 2, x^4 - x^3 + 15*x^2 + 17*x + 29, 23, 1, 1), ([13, 26, 52], 2, x^4 + 26*x^2 + 52, 36, 36, 2), ([53, 53, 53], 1, x^4 - x^3 + 7*x^2 + 43*x + 47, 13, 1, 1), ([61, 61, 549], 1, x^4 - x^3 + 8*x^2 - 42*x + 117, 39, 3, 1)]
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recip.bissonstreng.two_two_matrix(Zs, F)¶ Returns an n x n matrix A (where n=len(Zs)) with A[k,l] = 1 if Zs[k] and Zs[l] are (2,2)-isogenous, and A[k,l] = 0 otherwise
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recip.bissonstreng.visualize_orders(orders)¶ Returns an n x n matrix A (where n=len(Zs)) with A[k,l] = [OK:orders[k]] if orders[k]=orders[l], and A[k,l] = 0 otherwise