diff --git a/script/research/ec/pairing/3.1.2-classgroup.sage b/script/research/ec/pairing/3.1.2-classgroup.sage
index 61b69d66b..b262cad46 100644
--- a/script/research/ec/pairing/3.1.2-classgroup.sage
+++ b/script/research/ec/pairing/3.1.2-classgroup.sage
@@ -26,6 +26,7 @@ def degree(D):
 def neg(D):
     return tuple((-order, pnt) for order, pnt in D)
 
+# This means D1 ~ D2 because D1 - D2 ∈ Pic(E)
 D = D1 + neg(D2)
 assert degree(D) == 0
 assert dsum(D) == inf
diff --git a/script/research/ec/pairing/3.1.3-divlines.sage b/script/research/ec/pairing/3.1.3-divlines.sage
new file mode 100644
index 000000000..42772c176
--- /dev/null
+++ b/script/research/ec/pairing/3.1.3-divlines.sage
@@ -0,0 +1,31 @@
+q = 47
+Fq = GF(q)
+E = EllipticCurve(Fq, (0, 5))
+
+R.<x, y> = PolynomialRing(Fq)
+
+def add(P, Q):
+    Px, Py = P[0], P[1]
+    Qx, Qy = Q[0], Q[1]
+    # We don't handle this case yet :p
+    assert Px != Qx
+    gradient = (Qy - Py) / (Qx - Px)
+    intersect = Py - gradient * Px
+    f = y - gradient * x - intersect
+    assert f(Px, Py) == 0
+    assert f(Qx, Qy) == 0
+    R = P + Q
+    Rx, Ry = R[0], R[1]
+    assert f(Rx, -Ry) == 0
+    g = x - Rx
+    assert g(Rx, Ry) == g(Rx, -Ry) == 0
+    return f / g
+
+P = E(33, 9)
+Q = E(34, 39)
+# div(f) = [P] + [Q] - [P + Q] - [∞] ∈ Pic(E)
+#        = ([P] + [Q] - 2[∞]) - ([P + Q] - [∞])
+# => [P] + [Q] - 2[∞] ~ [P + Q] - [∞]
+f = add(P, Q)
+print(f)
+