89 lines
2.8 KiB
Python
89 lines
2.8 KiB
Python
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from sage.all import *
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from sage.misc.banner import require_version
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if not require_version(10, 5, print_message=True):
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exit('')
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from parameters import p, num_orders as num
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################################################################
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# Underlying theory:
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# - Ibukiyama, On maximal orders of division quaternion algebras with certain optimal embeddings
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# - https://ia.cr/2023/106 Lemma 10
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from sage.algebras.quatalg.quaternion_algebra import basis_for_quaternion_lattice
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bfql = lambda els: basis_for_quaternion_lattice(els, reverse=True)
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Quat1, (i,j,k) = QuaternionAlgebra(-1, -p).objgens()
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assert Quat1.discriminant() == p # ramifies correctly
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O0mat = matrix([list(g) for g in [Quat1(1), i, (i+j)/2, (1+k)/2]])
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O0 = Quat1.quaternion_order(list(O0mat))
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orders = [ (1, identity_matrix(QQ,4), O0mat, i, O0mat, vector((1,0,0,0))) ]
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q = ZZ(1)
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while len(orders) < num:
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q = next_prime(q)
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if q % 4 != 1: # restricting to q ≡ 1 (mod 4)
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continue
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Quatq, (ii,jj,kk) = QuaternionAlgebra(-q, -p).objgens()
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if Quatq.discriminant() != p: # ramifies incorrectly
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continue
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x, y = QuadraticForm(QQ, 2, [1,0,p]).solve(q)
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gamma = x + j*y
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assert gamma.reduced_norm() == q
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ims1 = [Quat1(1), i*gamma, j, k*gamma]
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assert ims1[1]**2 == -q
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assert ims1[2]**2 == -p
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assert ims1[1]*ims1[2] == ims1[3]
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assert ims1[2]*ims1[1] == -ims1[3]
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# (1,ii,jj,kk)->ims1 is an isomorphism Quatq->Quat1
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iso1q = ~matrix(map(list, ims1))
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r = min(map(ZZ, Mod(-p, 4*q).sqrt(all=True)))
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basq = [
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Quatq(1),
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ii,
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(1 + jj) / 2,
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(r + jj) * ii / 2 / q,
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]
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Oq = Quatq.quaternion_order(basq)
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assert Oq.discriminant() == p # is maximal
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mat1 = matrix(map(list, basq)) * ~iso1q
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O1 = Quat1.quaternion_order(list(mat1))
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assert O1.discriminant() == p # is maximal
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assert j in O1 # p-extremal
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# look for an odd connecting ideal
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I = O0 * O1
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I *= I.norm().denominator()
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assert I.is_integral()
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for v in IntegralLattice(I.gram_matrix()).enumerate_short_vectors():
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elt = sum(c*g for c,g in zip(v,I.basis()))
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if ZZ(elt.reduced_norm() / I.norm()) % 2:
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break
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I = I * (elt.conjugate() / I.norm())
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assert I.is_integral()
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assert I.norm() % 2
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assert I.left_order() == O0
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O1_ = I.right_order()
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assert O1_.unit_ideal() == elt * O1 * ~elt
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idl1 = matrix(map(list, I.basis()))
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# q
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# isomorphism from (-1,-p) algebra to (-q,-p) algebra
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# basis of maximal order O₁ in (-1,-p) algebra
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# element sqrt(-q) in O₁ in (-1,-p) algebra
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# basis of connecting ideal I from O₀ in (-1,-p) algebra
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# element γ such that I has right order γ O₁ γ^-1
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orders.append((q, iso1q, mat1, ims1[1], idl1, vector(elt)))
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