diff --git a/experimental/FTheoryTools/src/auxiliary.jl b/experimental/FTheoryTools/src/auxiliary.jl
index 127c3601715a..74d3c3b88e12 100644
--- a/experimental/FTheoryTools/src/auxiliary.jl
+++ b/experimental/FTheoryTools/src/auxiliary.jl
@@ -154,6 +154,7 @@ function _kodaira_type(id::MPolyIdeal{T}, f::T, g::T, d::T, ords::Tuple{Int64, I
   g_ord = ords[2]
   d_ord = ords[3]
     
+  # Check for cases where there cannot be Tate monodromy
   if d_ord == 0
     kod_type = "I_0"
   elseif d_ord == 1 && f_ord == 0 && g_ord == 0
@@ -169,6 +170,7 @@ function _kodaira_type(id::MPolyIdeal{T}, f::T, g::T, d::T, ords::Tuple{Int64, I
   elseif d_ord >= 12 && f_ord >= 4 && g_ord >= 6
     kod_type = "Non-minimal"
   else
+    # Create new ring with auxiliary variable to construct the monodromy polynomial
     R = parent(f)
     S, (_psi, ) = polynomial_ring(QQ, ["_psi"; [string(v) for v in gens(R)]], cached = false)
     ring_map = hom(R, S, gens(S)[2:end])
@@ -177,23 +179,39 @@ function _kodaira_type(id::MPolyIdeal{T}, f::T, g::T, d::T, ords::Tuple{Int64, I
     poly_d = ring_map(d)
     locus = ring_map(gens(id)[1])
     
+    # Compute monodromy polynomial and check factorization for remaining cases
     if f_ord == 0 && g_ord == 0
-      monodromy_poly = _psi^2 + divexact(evaluate(9 * poly_g, [locus], [0]), evaluate(2 * poly_f, [locus], [0]))
+      g_quotient = divrem(9 * poly_g, locus)[2]
+      f_quotient = divrem(2 * poly_f, locus)[2]
+      quotient_val = div(g_quotient, f_quotient)
+
+      monodromy_poly = _psi^2 + quotient_val
       kod_type = _string_from_factor_count(monodromy_poly, ["Non-split I_$d_ord", "Split I_$d_ord"])
     elseif d_ord == 4 && g_ord == 2 && f_ord >= 2
-      monodromy_poly = _psi^2 - evaluate(divexact(poly_g, locus^2), [locus], [0])
+      g_quotient = divrem(div(poly_g, locus^2), locus)[2]
+
+      monodromy_poly = _psi^2 - g_quotient
       kod_type = _string_from_factor_count(monodromy_poly, ["Non-split IV", "Split IV"])
     elseif d_ord == 6 && f_ord >= 2 && g_ord >= 3
-      monodromy_poly =  _psi^3 + _psi * evaluate(divexact(poly_f, locus^2), [locus], [0]) + evaluate(divexact(poly_g, locus^3), [locus], [0])
+      f_quotient = divrem(div(poly_f, locus^2), locus)[2]
+      g_quotient = divrem(div(poly_g, locus^3), locus)[2]
+      
+      monodromy_poly = _psi^3 + _psi * f_quotient + g_quotient
       kod_type = _string_from_factor_count(monodromy_poly, ["Non-split I^*_0", "Semi-split I^*_0", "Split I^*_0"])
-    elseif f_ord == 2 && g_ord == 3 && d_ord >= 7 && d_ord % 2 == 1
-      monodromy_poly = _psi^2 + divexact(evaluate(divexact(poly_d, locus^d_ord) * divexact(2 * poly_f, locus^2)^3, [locus], [0]), 4 * evaluate(divexact(9 * poly_g, locus^3), [locus], [0])^3)
-      kod_type = _string_from_factor_count(monodromy_poly, ["Non-split I^*_$(d_ord - 6)", "Split I^*_$(d_ord - 6)"])
-    elseif f_ord == 2 && g_ord == 3 && d_ord >= 8 && d_ord % 2 == 0
-      monodromy_poly = _psi^2 + divexact(evaluate(divexact(poly_d, locus^d_ord) * divexact(2 * poly_f, locus^2)^2, [locus], [0]), evaluate(divexact(9 * poly_g, locus^3), [locus], [0])^2)
+    elseif f_ord == 2 && g_ord == 3 && d_ord >= 7
+      d_quotient = div(poly_d, locus^d_ord)
+      f_quotient = div(2 * poly_f, locus^2)
+      g_quotient = div(9 * poly_g, locus^3)
+      num_quotient = divrem(d_quotient * f_quotient^(2 + d_ord % 2), locus)[2]
+      den_quotient = divrem(4 * g_quotient^(2 + d_ord % 2), locus)[2]
+      quotient_val = div(num_quotient, den_quotient)
+
+      monodromy_poly = _psi^2 + quotient_val
       kod_type = _string_from_factor_count(monodromy_poly, ["Non-split I^*_$(d_ord - 6)", "Split I^*_$(d_ord - 6)"])
     elseif d_ord == 8 && g_ord == 4 && f_ord >= 3
-      monodromy_poly = _psi^2 - evaluate(divexact(poly_g, locus^4), [locus], [0])
+      g_quotient = divrem(div(poly_g, locus^4), locus)[2]
+
+      monodromy_poly = _psi^2 - g_quotient
       kod_type = _string_from_factor_count(monodromy_poly, ["Non-split IV^*", "Split IV^*"])
     else
       kod_type = "Unrecognized"
diff --git a/experimental/FTheoryTools/test/tate_models.jl b/experimental/FTheoryTools/test/tate_models.jl
index 8e5881b14612..318afc21a4ea 100644
--- a/experimental/FTheoryTools/test/tate_models.jl
+++ b/experimental/FTheoryTools/test/tate_models.jl
@@ -98,43 +98,43 @@ end
 #############################################################
 
 # rings needed for constructions
-istar0_s_auxiliary_base_ring, (a1pp, a2pp, a3pp, a4pp, a6pp, vp, mp) = QQ["a1pp", "a2pp", "a3pp", "a4pp", "a6pp", "vp", "mp"];
-tate_auxiliary_base_ring, (a1p, a2p, a3p, a4p, a6p, v) = QQ["a1p", "a2p", "a3p", "a4p", "a6p", "v"];
+istar0_s_auxiliary_base_ring, (a1pp, a2pp, a3pp, a4pp, a6pp, vp, wp, mp) = QQ["a1pp", "a2pp", "a3pp", "a4pp", "a6pp", "vp", "wp", "mp"];
+tate_auxiliary_base_ring, (a1p, a2p, a3p, a4p, a6p, v, w) = QQ["a1p", "a2p", "a3p", "a4p", "a6p", "v", "w"];
 
 # construct Tate models over arbitrary base
-t_istar0_s = global_tate_model(istar0_s_auxiliary_base_ring, [1 2 3 4 6 0 2; -1 -2 -2 -3 -4 1 -1], 3, [a1pp * vp^1, mp * vp^1 + a2pp * vp^2, a3pp * vp^2, mp^2 * vp^2 + a4pp * vp^3, a6pp * vp^4]);
-t_i1 = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; 0 0 -1 -1 -1 1], 3, [a1p * v^0, a2p * v^0, a3p * v^1, a4p * v^1, a6p * v^1]);
-t_i2_ns = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; 0 0 -1 -1 -2 1], 3, [a1p * v^0, a2p * v^0, a3p * v^1, a4p * v^1, a6p * v^2]);
-t_i2_s = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; 0 -1 -1 -1 -2 1], 3, [a1p * v^0, a2p * v^1, a3p * v^1, a4p * v^1, a6p * v^2]);
-t_i3_ns = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; 0 0 -2 -2 -3 1], 3, [a1p * v^0, a2p * v^0, a3p * v^2, a4p * v^2, a6p * v^3]);
-t_i3_s = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; 0 -1 -1 -2 -3 1], 3, [a1p * v^0, a2p * v^1, a3p * v^1, a4p * v^2, a6p * v^3]);
-t_i4_ns = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; 0 0 -2 -2 -4 1], 3, [a1p * v^0, a2p * v^0, a3p * v^2, a4p * v^2, a6p * v^4]);
-t_i4_s = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; 0 -1 -2 -2 -4 1], 3, [a1p * v^0, a2p * v^1, a3p * v^2, a4p * v^2, a6p * v^4]);
-t_i5_ns = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; 0 0 -3 -3 -5 1], 3, [a1p * v^0, a2p * v^0, a3p * v^3, a4p * v^3, a6p * v^5]);
+t_istar0_s = global_tate_model(istar0_s_auxiliary_base_ring, [1 2 3 4 6 0 2; -1 -2 -2 -3 -4 1 -1], 3, [a1pp * (vp^2 + wp)^1, mp * (vp^2 + wp)^1 + a2pp * (vp^2 + wp)^2, a3pp * (vp^2 + wp)^2, mp^2 * (vp^2 + wp)^2 + a4pp * (vp^2 + wp)^3, a6pp * (vp^2 + wp)^4]);
+t_i1 = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; 0 0 -1 -1 -1 1], 3, [a1p * (v^2 + w)^0, a2p * (v^2 + w)^0, a3p * (v^2 + w)^1, a4p * (v^2 + w)^1, a6p * (v^2 + w)^1]);
+t_i2_ns = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; 0 0 -1 -1 -2 1], 3, [a1p * (v^2 + w)^0, a2p * (v^2 + w)^0, a3p * (v^2 + w)^1, a4p * (v^2 + w)^1, a6p * (v^2 + w)^2]);
+t_i2_s = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; 0 -1 -1 -1 -2 1], 3, [a1p * (v^2 + w)^0, a2p * (v^2 + w)^1, a3p * (v^2 + w)^1, a4p * (v^2 + w)^1, a6p * (v^2 + w)^2]);
+t_i3_ns = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; 0 0 -2 -2 -3 1], 3, [a1p * (v^2 + w)^0, a2p * (v^2 + w)^0, a3p * (v^2 + w)^2, a4p * (v^2 + w)^2, a6p * (v^2 + w)^3]);
+t_i3_s = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; 0 -1 -1 -2 -3 1], 3, [a1p * (v^2 + w)^0, a2p * (v^2 + w)^1, a3p * (v^2 + w)^1, a4p * (v^2 + w)^2, a6p * (v^2 + w)^3]);
+t_i4_ns = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; 0 0 -2 -2 -4 1], 3, [a1p * (v^2 + w)^0, a2p * (v^2 + w)^0, a3p * (v^2 + w)^2, a4p * (v^2 + w)^2, a6p * (v^2 + w)^4]);
+t_i4_s = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; 0 -1 -2 -2 -4 1], 3, [a1p * (v^2 + w)^0, a2p * (v^2 + w)^1, a3p * (v^2 + w)^2, a4p * (v^2 + w)^2, a6p * (v^2 + w)^4]);
+t_i5_ns = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; 0 0 -3 -3 -5 1], 3, [a1p * (v^2 + w)^0, a2p * (v^2 + w)^0, a3p * (v^2 + w)^3, a4p * (v^2 + w)^3, a6p * (v^2 + w)^5]);
 t_i5_s = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; 0 -1 -2 -3 -5 1], 3, [a1p * v^0, a2p * v^1, a3p * v^2, a4p * v^3, a6p * v^5]);
-t_i6_ns = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; 0 0 -3 -3 -6 1], 3, [a1p * v^0, a2p * v^0, a3p * v^3, a4p * v^3, a6p * v^6]);
-t_i6_s = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; 0 -1 -3 -3 -6 1], 3, [a1p * v^0, a2p * v^1, a3p * v^3, a4p * v^3, a6p * v^6]);
-t_i7_ns = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; 0 0 -4 -4 -7 1], 3, [a1p * v^0, a2p * v^0, a3p * v^4, a4p * v^4, a6p * v^7]);
-t_i7_s = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; 0 -1 -3 -4 -7 1], 3, [a1p * v^0, a2p * v^1, a3p * v^3, a4p * v^4, a6p * v^7]);
-t_ii = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; -1 -1 -1 -1 -1 1], 3, [a1p * v^1, a2p * v^1, a3p * v^1, a4p * v^1, a6p * v^1]);
-t_iii = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; -1 -1 -1 -1 -2 1], 3, [a1p * v^1, a2p * v^1, a3p * v^1, a4p * v^1, a6p * v^2]);
-t_iv_ns = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; -1 -1 -1 -2 -2 1], 3, [a1p * v^1, a2p * v^1, a3p * v^1, a4p * v^2, a6p * v^2]);
-t_iv_s = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; -1 -1 -1 -2 -3 1], 3, [a1p * v^1, a2p * v^1, a3p * v^1, a4p * v^2, a6p * v^3]);
-t_istar0_ns = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; -1 -1 -2 -2 -3 1], 3, [a1p * v^1, a2p * v^1, a3p * v^2, a4p * v^2, a6p * v^3]);
-t_istar0_ss = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; -1 -1 -2 -2 -4 1], 3, [a1p * v^1, a2p * v^1, a3p * v^2, a4p * v^2, a6p * v^4]);
-t_istar1_ns = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; -1 -1 -2 -3 -4 1], 3, [a1p * v^1, a2p * v^1, a3p * v^2, a4p * v^3, a6p * v^4]);
-t_istar1_s = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; -1 -1 -2 -3 -5 1], 3, [a1p * v^1, a2p * v^1, a3p * v^2, a4p * v^3, a6p * v^5]);
-t_istar2_ns = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; -1 -1 -3 -3 -5 1], 3, [a1p * v^1, a2p * v^1, a3p * v^3, a4p * v^3, a6p * v^5]);
-t_istar2_s = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; -1 -1 -3 -3 -6 1], 3, [a1p * v^1, a2p * v^1, a3p * v^3, a4p * v^3, a6p * v^6]);
-t_istar3_ns = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; -1 -1 -3 -4 -6 1], 3, [a1p * v^1, a2p * v^1, a3p * v^3, a4p * v^4, a6p * v^6]);
-t_istar3_s = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; -1 -1 -3 -4 -7 1], 3, [a1p * v^1, a2p * v^1, a3p * v^3, a4p * v^4, a6p * v^7]);
-t_istar4_ns = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; -1 -1 -4 -4 -7 1], 3, [a1p * v^1, a2p * v^1, a3p * v^4, a4p * v^4, a6p * v^7]);
-t_istar4_s = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; -1 -1 -4 -4 -8 1], 3, [a1p * v^1, a2p * v^1, a3p * v^4, a4p * v^4, a6p * v^8]);
-t_ivstar_ns = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; -1 -2 -2 -3 -4 1], 3, [a1p * v^1, a2p * v^2, a3p * v^2, a4p * v^3, a6p * v^4]);
-t_ivstar_s = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; -1 -2 -2 -3 -5 1], 3, [a1p * v^1, a2p * v^2, a3p * v^2, a4p * v^3, a6p * v^5]);
-t_iiistar = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; -1 -2 -3 -3 -5 1], 3, [a1p * v^1, a2p * v^2, a3p * v^3, a4p * v^3, a6p * v^5]);
-t_iistar = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; -1 -2 -3 -4 -5 1], 3, [a1p * v^1, a2p * v^2, a3p * v^3, a4p * v^4, a6p * v^5]);
-t_nm = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; -1 -2 -3 -4 -6 1], 3, [a1p * v^1, a2p * v^2, a3p * v^3, a4p * v^4, a6p * v^6]);
+t_i6_ns = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; 0 0 -3 -3 -6 1], 3, [a1p * (v^2 + w)^0, a2p * (v^2 + w)^0, a3p * (v^2 + w)^3, a4p * (v^2 + w)^3, a6p * (v^2 + w)^6]);
+t_i6_s = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; 0 -1 -3 -3 -6 1], 3, [a1p * (v^2 + w)^0, a2p * (v^2 + w)^1, a3p * (v^2 + w)^3, a4p * (v^2 + w)^3, a6p * (v^2 + w)^6]);
+t_i7_ns = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; 0 0 -4 -4 -7 1], 3, [a1p * (v^2 + w)^0, a2p * (v^2 + w)^0, a3p * (v^2 + w)^4, a4p * (v^2 + w)^4, a6p * (v^2 + w)^7]);
+t_i7_s = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; 0 -1 -3 -4 -7 1], 3, [a1p * (v^2 + w)^0, a2p * (v^2 + w)^1, a3p * (v^2 + w)^3, a4p * (v^2 + w)^4, a6p * (v^2 + w)^7]);
+t_ii = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; -1 -1 -1 -1 -1 1], 3, [a1p * (v^2 + w)^1, a2p * (v^2 + w)^1, a3p * (v^2 + w)^1, a4p * (v^2 + w)^1, a6p * (v^2 + w)^1]);
+t_iii = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; -1 -1 -1 -1 -2 1], 3, [a1p * (v^2 + w)^1, a2p * (v^2 + w)^1, a3p * (v^2 + w)^1, a4p * (v^2 + w)^1, a6p * (v^2 + w)^2]);
+t_iv_ns = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; -1 -1 -1 -2 -2 1], 3, [a1p * (v^2 + w)^1, a2p * (v^2 + w)^1, a3p * (v^2 + w)^1, a4p * (v^2 + w)^2, a6p * (v^2 + w)^2]);
+t_iv_s = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; -1 -1 -1 -2 -3 1], 3, [a1p * (v^2 + w)^1, a2p * (v^2 + w)^1, a3p * (v^2 + w)^1, a4p * (v^2 + w)^2, a6p * (v^2 + w)^3]);
+t_istar0_ns = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; -1 -1 -2 -2 -3 1], 3, [a1p * (v^2 + w)^1, a2p * (v^2 + w)^1, a3p * (v^2 + w)^2, a4p * (v^2 + w)^2, a6p * (v^2 + w)^3]);
+t_istar0_ss = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; -1 -1 -2 -2 -4 1], 3, [a1p * (v^2 + w)^1, a2p * (v^2 + w)^1, a3p * (v^2 + w)^2, a4p * (v^2 + w)^2, a6p * (v^2 + w)^4]);
+t_istar1_ns = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; -1 -1 -2 -3 -4 1], 3, [a1p * (v^2 + w)^1, a2p * (v^2 + w)^1, a3p * (v^2 + w)^2, a4p * (v^2 + w)^3, a6p * (v^2 + w)^4]);
+t_istar1_s = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; -1 -1 -2 -3 -5 1], 3, [a1p * (v^2 + w)^1, a2p * (v^2 + w)^1, a3p * (v^2 + w)^2, a4p * (v^2 + w)^3, a6p * (v^2 + w)^5]);
+t_istar2_ns = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; -1 -1 -3 -3 -5 1], 3, [a1p * (v^2 + w)^1, a2p * (v^2 + w)^1, a3p * (v^2 + w)^3, a4p * (v^2 + w)^3, a6p * (v^2 + w)^5]);
+t_istar2_s = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; -1 -1 -3 -3 -6 1], 3, [a1p * (v^2 + w)^1, a2p * (v^2 + w)^1, a3p * (v^2 + w)^3, a4p * (v^2 + w)^3, a6p * (v^2 + w)^6]);
+t_istar3_ns = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; -1 -1 -3 -4 -6 1], 3, [a1p * (v^2 + w)^1, a2p * (v^2 + w)^1, a3p * (v^2 + w)^3, a4p * (v^2 + w)^4, a6p * (v^2 + w)^6]);
+t_istar3_s = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; -1 -1 -3 -4 -7 1], 3, [a1p * (v^2 + w)^1, a2p * (v^2 + w)^1, a3p * (v^2 + w)^3, a4p * (v^2 + w)^4, a6p * (v^2 + w)^7]);
+t_istar4_ns = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; -1 -1 -4 -4 -7 1], 3, [a1p * (v^2 + w)^1, a2p * (v^2 + w)^1, a3p * (v^2 + w)^4, a4p * (v^2 + w)^4, a6p * (v^2 + w)^7]);
+t_istar4_s = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; -1 -1 -4 -4 -8 1], 3, [a1p * (v^2 + w)^1, a2p * (v^2 + w)^1, a3p * (v^2 + w)^4, a4p * (v^2 + w)^4, a6p * (v^2 + w)^8]);
+t_ivstar_ns = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; -1 -2 -2 -3 -4 1], 3, [a1p * (v^2 + w)^1, a2p * (v^2 + w)^2, a3p * (v^2 + w)^2, a4p * (v^2 + w)^3, a6p * (v^2 + w)^4]);
+t_ivstar_s = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; -1 -2 -2 -3 -5 1], 3, [a1p * (v^2 + w)^1, a2p * (v^2 + w)^2, a3p * (v^2 + w)^2, a4p * (v^2 + w)^3, a6p * (v^2 + w)^5]);
+t_iiistar = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; -1 -2 -3 -3 -5 1], 3, [a1p * (v^2 + w)^1, a2p * (v^2 + w)^2, a3p * (v^2 + w)^3, a4p * (v^2 + w)^3, a6p * (v^2 + w)^5]);
+t_iistar = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; -1 -2 -3 -4 -5 1], 3, [a1p * (v^2 + w)^1, a2p * (v^2 + w)^2, a3p * (v^2 + w)^3, a4p * (v^2 + w)^4, a6p * (v^2 + w)^5]);
+t_nm = global_tate_model(tate_auxiliary_base_ring, [1 2 3 4 6 0; -1 -2 -3 -4 -6 1], 3, [a1p * (v^2 + w)^1, a2p * (v^2 + w)^2, a3p * (v^2 + w)^3, a4p * (v^2 + w)^4, a6p * (v^2 + w)^6]);
 
 @testset "Attributes of global Tate models over generic base space" begin
   @test parent(tate_section_a1(t_i5_s)) == coordinate_ring(base_space(t_i5_s))
@@ -233,7 +233,7 @@ end
   irr_i5_s = irrelevant_ideal(tas);
   sri_i5_s = stanley_reisner_ideal(tas);
   lin_i5_s = ideal_of_linear_relations(tas);
-  id_fin,  = _blowup_global_sequence(id_i5_s, [[7, 8, 6], [2, 3, 1], [3, 4], [2, 4]], irr_i5_s, sri_i5_s, lin_i5_s)
+  id_fin,  = _blowup_global_sequence(id_i5_s, [[8, 9, 6], [2, 3, 1], [3, 4], [2, 4]], irr_i5_s, sri_i5_s, lin_i5_s)
   @test string(gens(id_fin)[end]) == "-b_4_1*b_2_1*a1p*z - b_4_1*b_2_2 - b_4_1*b_2_3*b_1_3^2*a3p*z^3 + b_4_2*b_3_2*b_2_1^2*b_1_1 + b_4_2*b_3_2*b_2_1^2*b_1_3*a2p*z^2 + b_4_2*b_3_2*b_2_1*b_2_3*b_1_3^3*a4p*z^4 + b_4_2*b_3_2*b_2_3^2*b_1_3^5*a6p*z^6"
 end