diff --git a/tests/playground_models.sage b/tests/playground_models.sage
new file mode 100644
index 0000000..d4c624b
--- /dev/null
+++ b/tests/playground_models.sage
@@ -0,0 +1,104 @@
+# Entries are (id, factory, expected_cy_dimension, expected_no_divs, expected_disc,
+# expected_intersection_numbers_CY), where expected_disc is a list of
+# (str(discriminant_factor), codimension) pairs as returned by ToricPolytope.disc().
+
+PLAYGROUND_MODEL_FACTORIES = [
+    (
+        "K3_two_parameter_family",
+        lambda: ToricPolytopeProjectiveSpace([1, 1, 2, 4], model_name="K3_two_parameter_family"),
+        2,  # cy_dimension
+        2,  # no_divs
+        [("z2 - 1/4", 2), ("4096*z1^2*(4*z2 - 1) + 128*z1 - 1", 0)],
+        {(0, 0): 4, (0, 1): 2, (1, 1): 0},
+    ),
+    (
+        "CY3_two_parameter_family",
+        lambda: ToricPolytopeProjectiveSpace([1, 1, 2, 2, 2], model_name="CY3_two_parameter_family"),
+        3,
+        2,
+        [("z2 - 1/4", 3), ("65536*z1^2*(4*z2 - 1) + 512*z1 - 1", 0)],
+        {(0, 0, 0): 8, (0, 0, 1): 4, (0, 1, 1): 0, (1, 1, 1): 0},
+    ),
+    (
+        "CY4_two_parameter_family",
+        lambda: ToricPolytopeProjectiveSpace([1, 1, 1, 1, 8, 12], model_name="CY4_two_parameter_family"),
+        4,
+        2,
+        [
+            ("z2 - 1/256", 2),
+            ("34828517376*z1^4*(256*z2 - 1) + 322486272*z1^3 - 1119744*z1^2 + 1728*z1 - 1", 0),
+        ],
+        {(0, 0, 0, 0): 64, (0, 0, 0, 1): 16, (0, 0, 1, 1): 4, (0, 1, 1, 1): 1, (1, 1, 1, 1): 0},
+    ),
+    (
+        "CICY3_one_parameter",
+        lambda: ToricPolytopeCICY([[3, 3]], model_name="CICY3_one_parameter"),
+        3,
+        1,
+        [("z1 - 1/46656", 0)],
+        {(0, 0, 0): 9},
+    ),
+    (
+        "CICY3_two_parameter",
+        lambda: ToricPolytopeCICY([[3], [3]], model_name="CICY3_two_parameter"),
+        3,
+        2,
+        [
+            (
+                "-19683*z1^3 - 2187*(27*z1 + 1)*z2^2 - 19683*z2^3 - 2187*z1^2 "
+                "- 81*(729*z1^2 - 189*z1 + 1)*z2 - 81*z1 - 1",
+                0,
+            )
+        ],
+        {(0, 0, 0): 0, (0, 0, 1): 3, (0, 1, 1): 3, (1, 1, 1): 0},
+    ),
+    (
+        "CICY5_two_parameter",
+        lambda: ToricPolytopeCICY([[6, 1], [0, 2]], model_name="CICY5_two_parameter"),
+        5,
+        2,
+        [
+            (
+                "16384*z2^7 - 28672*z2^6 + 21504*z2^5 - 448*(1647086*z1 - 5)*z2^3 - 8960*z2^4 "
+                "- 112*(8235430*z1 + 3)*z2^2 - 678223072849*z1^2 - 28*(4941258*z1 - 1)*z2 - 1647086*z1 - 1",
+                0,
+            )
+        ],
+        {
+            (0, 0, 0, 0, 0): 6,
+            (0, 0, 0, 0, 1): 12,
+            (0, 0, 0, 1, 1): 0,
+            (0, 0, 1, 1, 1): 0,
+            (0, 1, 1, 1, 1): 0,
+            (1, 1, 1, 1, 1): 0,
+        },
+    ),
+    (
+        "CICY3_two_parameter_manual",
+        lambda: ToricPolytope(
+            [
+                [1, 0, 0, 0, 0, 0],
+                [0, 1, 0, 0, 0, 0],
+                [0, 0, 1, 0, 0, 0],
+                [0, 0, 0, 1, 0, 0],
+                [0, 0, 0, 0, 1, 0],
+                [0, 0, 0, 0, 0, 1],
+                [-1, -1, 0, 0, 0, 0],
+                [0, 0, -1, -1, -1, -1],
+            ],
+            nef_partition=[[0, 1, 2, 3], [4, 5, 6, 7]],
+            model_name="CICY3_two_parameter_manual",
+        ),
+        4,
+        2,
+        [
+            (
+                "-14348907*z1^5 - 2657205*z1^4 - 196830*z1^3 - 29296875*(270*z1 + 1)*z2^2 "
+                "- 30517578125*z2^3 - 7290*z1^2 + 9375*(98415*z1^3 - 32805*z1^2 + 810*z1 - 1)*z2 "
+                "- 135*z1 - 1",
+                0,
+            )
+        ],
+        {(0, 0, 0, 0): 0, (0, 0, 0, 1): 0, (0, 0, 1, 1): 6, (0, 1, 1, 1): 8, (1, 1, 1, 1): 2},
+    ),
+]
diff --git a/tests/test_smoke.py b/tests/test_smoke.py
deleted file mode 100644
index 72d7923..0000000
--- a/tests/test_smoke.py
+++ /dev/null
@@ -1,32 +0,0 @@
-import os
-
-import pytest
-from sage.all import *  # noqa: F401,F403
-
-load(os.path.join(os.path.dirname(__file__), "..", "sage", "toric_topdata.sage"))
-
-
-def test_polytope_rejects_too_few_points():
-    with pytest.raises(ValueError):
-        Polytope([[0, 0], [1, 0]])  # 2 points in dimension 2: not enough
-
-
-def test_mirror_quintic_topdata():
-    # Vertices e_1, ..., e_4, -e_1-...-e_4 -- the worked example from
-    # ToricPolytope's own docstring.
-    points = [
-        [1, 0, 0, 0],
-        [0, 1, 0, 0],
-        [0, 0, 1, 0],
-        [0, 0, 0, 1],
-        [-1, -1, -1, -1],
-    ]
-
-    polytope = ToricPolytope(points, model_name="ci_smoke_test_quintic")
-    assert polytope.dimension == 4
-    assert len(polytope.triangulations) >= 1
-
-    polytope.topdata()
-    assert polytope.cy_dimension == 3
-    assert polytope.no_divs == 1  # P^4 has Picard rank 1
-    assert polytope.intersection_numbers_CY == {(0, 0, 0): 5}  # classical quintic self-intersection
diff --git a/tests/topdata_tests.py b/tests/topdata_tests.py
new file mode 100644
index 0000000..94e950f
--- /dev/null
+++ b/tests/topdata_tests.py
@@ -0,0 +1,34 @@
+import os
+
+import pytest
+from sage.all import *  # noqa: F401,F403
+from sage.geometry.lattice_polytope import set_palp_dimension
+
+load(os.path.join(os.path.dirname(__file__), "..", "sage", "toric_topdata.sage"))
+load(os.path.join(os.path.dirname(__file__), "playground_models.sage"))
+
+set_palp_dimension(11)
+
+PLAYGROUND_MODELS = [pytest.param(*row[1:], id=row[0]) for row in PLAYGROUND_MODEL_FACTORIES]  # noqa: F821
+
+
+@pytest.mark.parametrize(
+    "make_model, expected_cy_dimension, expected_no_divs, expected_disc, expected_intersection_numbers_CY",
+    PLAYGROUND_MODELS,
+)
+def test_playground_models_disc_and_topdata(
+    make_model,
+    expected_cy_dimension,
+    expected_no_divs,
+    expected_disc,
+    expected_intersection_numbers_CY,
+):
+    polytope = make_model()
+
+    discriminants = polytope.disc()
+    assert [(str(factor), codim) for factor, codim in discriminants] == expected_disc
+
+    polytope.topdata()
+    assert polytope.cy_dimension == expected_cy_dimension
+    assert polytope.no_divs == expected_no_divs
+    assert polytope.intersection_numbers_CY == expected_intersection_numbers_CY