Finally added the old method of embed_order, as embed_order_legacy.

Author: mmasdeu

arithgroup.py

@@ -824,6 +824,62 @@ def embed_order(self,p,K,prec,orientation = None, use_magma = True,outfile = Non
         gamma.set_immutable()
         return self(gamma), v0(tau)
 
+    def embed_order_legacy(self,p,K,prec,outfile = None,return_all = False):
+        r'''
+        '''
+        from limits import _find_initial_embedding_list,find_optimal_embeddings,order_and_unit, find_the_unit_of
+
+        verbose('Computing quadratic embedding to precision %s'%prec)
+        mu = find_optimal_embeddings(K,use_magma = True, extra_conductor = 1)[-1]
+        verbose('Finding module generators')
+        w = module_generators(K)[1]
+        verbose('Done')
+        w_minpoly = w.minpoly().change_ring(Qp(p,prec))
+        Cp = Qp(p,prec).extension(w_minpoly,names = 'g')
+        wl = w.list()
+        assert len(wl) == 2
+        r0 = -wl[0]/wl[1]
+        r1 = 1/wl[1]
+        assert r0 + r1 * w == K.gen()
+        padic_Kgen = Cp(r0)+Cp(r1)*Cp.gen()
+        try:
+            fwrite('d_K = %s, h_K = %s, h_K^- = %s'%(K.discriminant(),K.class_number(),len(K.narrow_class_group())),outfile)
+        except NotImplementedError: pass
+        fwrite('w_K satisfies: %s'%w.minpoly(),outfile)
+        mu = r0 + r1*mu
+        assert K.gen(0).trace() == mu.trace() and K.gen(0).norm() == mu.determinant()
+        iotap = self.get_embedding(prec)
+        a,b,c,d = iotap(mu).list()
+        X = PolynomialRing(Cp,names = 'X').gen()
+        tau1 = (Cp(a-d) + 2*padic_Kgen)/Cp(2*c)
+        tau2 = (Cp(a-d) - 2*padic_Kgen)/Cp(2*c)
+        assert (Cp(c)*tau1**2 + Cp(d-a)*tau1-Cp(b)) == 0
+        assert (Cp(c)*tau2**2 + Cp(d-a)*tau2-Cp(b)) == 0
+        u = find_the_unit_of(self.F,K)
+        gammalst = u.list()
+        assert len(gammalst) == 2
+        gammaquatrep = self.B(gammalst[0]) + self.B(gammalst[1]) * mu
+        verbose('gammaquatrep trd = %s and nrd = %s'%(gammaquatrep.trace(),gammaquatrep.determinant()))
+        verbose('u trace = %s and unorm = %s'%(u.trace(),u.norm()))
+        assert gammaquatrep.trace() == u.trace() and gammaquatrep.determinant() == u.norm()
+        gammaq = gammaquatrep
+        while True:
+            try:
+                gamma = self(gammaq)
+                break
+            except ValueError:
+                gammaq *= gammaquatrep
+        a, b, c, d = iotap(gamma.quaternion_rep).list()
+        assert (c*tau1**2 + (d-a)*tau1 - b) == 0
+        fwrite('\cO_K to R_0 given by w_K |-> %s'%mu,outfile)
+        fwrite('gamma_psi = %s'%gamma,outfile)
+        fwrite('tau_psi = %s'%tau1,outfile)
+        fwrite('(where g satisfies: %s)'%w.minpoly(),outfile)
+        if return_all:
+            return gamma, tau1, tau2
+        else:
+            return gamma, tau1
+
     def check_word(self,delta,wd):
         tmp = self.B(1)
         for i,a in wd: