..nodoctest
Finding class invariants¶
x.__init__(...) initializes x; see help(type(x)) for signature
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recip.find_class_invariant.a_to_mu(Psi, A)¶ Given an ideal A and CM-type Psi, returns mu with mu*mubar in QQ and N_Psi(A) = (mu) if mu exists, and None otherwise
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recip.find_class_invariant.list_group(gens)¶ Given a set of elements of a finite group, lists all elements of the group generated by the given elements.
NOTE: Use
group_generators_to_list()instead, as it is better. The function list_group will be removed in a later version.EXAMPLES:
The matrices M and N were output by an earlier version of reciprocity_map_image. The outputs are equivalent, as the following test shows.:
sage: from recip import * sage: k = CM_Field((x^2+5)^2-4*5) sage: Z = k.period_matrices_iter().next() sage: gens = reciprocity_map_image(Z, 6) sage: lst = list_group(gens) # long time Warning: the function list_group will be removed in a later version, use group_generators_to_list instead sage: M = Matrix(Zmod(6), [[3,1,2,5],[0,2,5,1],[4,2,4,5],[1,4,4,4]]) sage: M in lst # long time, not tested, apparently wrong?? True sage: N = Matrix(Zmod(6), [[3,0,4,0],[0,5,0,4],[2,4,3,2],[0,2,2,3]]) sage: N in lst # long time, not tested, apparently wrong?? True sage: lst = list_group([M,N]) # long time Warning: the function list_group will be removed in a later version, use group_generators_to_list instead sage: all([g.matrix() in lst for g in gens]) # long time, not tested, apparently wrong?? True
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recip.find_class_invariant.make_big_zero(n, big)¶ Returns \(0\) if \(n==big\) and returns \(n\) otherwise. We need this because permutation groups don’t handle 0 well.
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recip.find_class_invariant.make_zero_big(n, big)¶ Returns \(big\) if \(n==0\) and returns \(n\) otherwise. We need this because permutation groups don’t handle 0 well.
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recip.find_class_invariant.principal_type_norms(Psi, modulus)¶ Returns a set of generators of the image of (H(1) cap I(modulus)) / H(modulus) in (Or / modulus)^* / O^* under the type norm of Psi.
Here I(N) is the group of ideals of the domain of Psi, coprime to N, and H(N) is the subgroup of ideals A such that the type norm of A is principal and generated by mu with mu*mubar in QQ.
Or is the maximal order of the reflex field.
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recip.find_class_invariant.reciprocity_map_image(Z, level, modulus=None)¶ Given a CM period matrix Z, returns Z.epsilon(b) mod level for a set of generators A of the group (H(1) cap I(modulus)) / H(modulus).
Here I(N) is the group of ideals of the reflex field coprime to N, and H(N) is the subgroup of ideals A such that the reflex type norm of A is principal and generated by mu with mu*mubar in QQ.
The b in Z.epsilon(b) is b=mu with mu as in the definition of H(1).
If modulus is None, then take modulus=level
EXAMPLE:
sage: from recip import * sage: k = CM_Field((x^2+5)^2-4*5) sage: Z = k.one_period_matrix(); Z Period Matrix [-0.30901699437495? + 0.95105651629515?*I -0.50000000000000? + 0.36327126400268?*I] [-0.50000000000000? + 0.36327126400268?*I 0.30901699437495? + 0.95105651629515?*I] sage: reciprocity_map_image(Z, 6) # not tested, is this answer correct? [[1 1 4 1] [2 4 1 5] [2 2 0 3] [1 2 4 2], [3 2 2 0] [2 3 0 2] [4 0 3 0] [2 4 0 5]] sage: gammas = reciprocity_map_image(Z, 8) sage: len(gammas) 4 sage: gammas[0] # not tested, is this answer correct? [0 2 1 6] [0 1 6 7] [7 1 2 1] [2 7 7 2]
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recip.find_class_invariant.table(gens, den, select_rows=None)¶ Returns a table showing the action of gens on the theta’s with characteristics in 1/den.
EXAMPLE:
sage: from recip import * sage: gens = [GSp_element([[5,0,7,1],[6,6,0,7],[3,6,6,1],[6,1,7,4]],ring=Zmod(8))] sage: table(gens, 2, select_rows=[0,1,4,6,8]) [ 0| 1] [-------------+-------------] [ t0| -t6] [ t1| t0] [ t4| (zeta8)*t1] [ t6|(-zeta8^3)*t8] [ t8|(-zeta8^2)*t4] [ (zeta8)| (zeta8^3)]
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recip.find_class_invariant.visualize(gens, den, select_rows=None)¶ Gives a visual representation of the action of gens on the theta’s of characteristic in ZZ/den.
EXAMPLE:
sage: from recip import * sage: M = [] sage: M.append([[5,0,7,1],[6,6,0,7],[3,6,6,1],[6,1,7,4]]) sage: M.append([[1,6,4,2],[2,3,6,4],[0,4,5,2],[4,4,6,7]]) sage: M.append([[7,0,6,4],[2,1,0,6],[0,2,1,2],[2,2,4,3]]) sage: M.append([[5,0,4,0],[4,1,0,4],[0,4,1,4],[4,4,0,5]]) sage: gens = [GSp_element(m, ring=Zmod(8)) for m in M] sage: visualize(gens, 2) On the 8-th powers, the action is (note that 16 means 0): 1: (1,16,6,8,4)(2,15,12,3,9) 2: () 3: () 4: () Permutation Group with generators [(), (1,16,6,8,4)(2,15,12,3,9)] of order 5 The action on the orbit [0, 1, 4, 6, 8] is as follows [ 0| 1 2 3 4] [-------------+-------------------------------------------------------] [ t0| -t6 t0 t0 t0] [ t1| t0 t1 -t1 t1] [ t4| (zeta8)*t1 t4 (zeta8^2)*t4 -t4] [ t6|(-zeta8^3)*t8 t6 -t6 t6] [ t8|(-zeta8^2)*t4 t8 (-zeta8^2)*t8 -t8] [ (zeta8)| (zeta8^3) (zeta8) (-zeta8^3) (-zeta8)] The action on the orbit [2, 3, 9, 12, 15] is as follows [ 0| 1 2 3 4] [-------------+-------------------------------------------------------] [ t2| t15 -t2 (zeta8^2)*t2 -t2] [ t3| (zeta8^2)*t9 -t3 (zeta8^2)*t3 -t3] [ t9|(-zeta8^2)*t2 -t9 (-zeta8^2)*t9 -t9] [ t12| (zeta8^3)*t3 -t12 t12 t12] [ t15|(zeta8^3)*t12 -t15 (zeta8^2)*t15 -t15] [ (zeta8)| (zeta8^3) (zeta8) (-zeta8^3) (-zeta8)]