bug: fix _find_quaternion_algebra (thofma#1521)

- if P^n is principal, so is P^(n mod ord(P)), so we can always
  assume that exponents are nonnegative by reducing modulo the
  order
- closes thofma#1512

src/QuadForm/Misc.jl

@@ -889,6 +889,7 @@ function _find_quaternion_algebra(b, P, I)
     Cl, mCl = class_group(R)
     A = abelian_group(fill(0, length(__P)))
     hh = hom(A, Cl, [mCl\(p) for p in __P])
+    _orders = [order(mCl\(p)) for p in __P]
     S, mS = image(hh, false)
     Q, mQ = quo(Cl, [mS(S[i]) for i in 1:ngens(S)])
 
@@ -901,7 +902,7 @@ function _find_quaternion_algebra(b, P, I)
         end
         o = order(mQ(mCl\(q)))
         c = -(hh\(o * (mCl\(q))))
-        fl, x = is_principal_with_data(q * prod(__P[i]^Int(c.coeff[i]) for i in 1:length(__P)))
+        fl, x = is_principal_with_data(q * prod(__P[i]^mod(Int(c.coeff[i]), Int(_orders[i])) for i in 1:length(__P)))
         @assert fl
         v = sign_vector(elem_in_nf(x))
         if rank(M) == rank(vcat(M, v + target))

test/QuadForm/Herm/GenusRep.jl

@@ -149,4 +149,24 @@
   H = lattice(hermitian_space(E, 1))
   @test length(genus_representatives(H)) == 1
 
+  let
+    # 1512
+    Qx, x = polynomial_ring(FlintQQ, "x")
+    f = x^3 - 6*x^2 - 4*x + 23
+    K, a = number_field(f, "a", cached = false)
+    Kt, t = polynomial_ring(K, "t")
+    g = t^2 - a*t + 1
+    E, b = number_field(g, "b", cached = false);
+    S = unique([restrict(r, K) for r in filter(!is_real, infinite_places(E)) if is_real(restrict(r, K))]);
+    sort!(S, lt=(p,q) -> isless(real(embedding(p).r), real(embedding(q).r)));
+    vals = Int[1, 1];
+    sig = Dict(S[i] => vals[i] for i in 1:2);
+    OK = maximal_order(K);
+    ps = AbsSimpleNumFieldOrderIdeal[ideal(OK, v) for v in Vector{AbsSimpleNumFieldOrderElem}[map(OK, [2, a^2 + a + 1]), map(OK, [2, a + 3]), map(OK, [11, a + 6]), map(OK, [239, a + 174]), map(OK, [1487, a + 881])]];
+    datas = [[(0, 1, 1)], [(0, 1, 1)], [(-1, 1, -1)], [(3, 1, -1)], [(-1, 1, 1)]];
+    lgs = HermLocalGenus{typeof(E), AbsSimpleNumFieldOrderIdeal}[genus(HermLat, E, ps[i], datas[i]) for i in 1:5];
+    G = HermGenus(E, 1, lgs, sig)
+    GG = representative(G)
+    @test GG in G
+  end
 end