diff --git a/src/pasta/curves.rs b/src/pasta/curves.rs
index 57e5c2c..94fcc32 100644
--- a/src/pasta/curves.rs
+++ b/src/pasta/curves.rs
@@ -800,15 +800,14 @@ macro_rules! impl_projective_curve_specific {
             let yy = self.y.square();
             let a = yy.square();
             let zz = self.z.square();
-            let s = (self.x + yy).square() - xx - a;
-            let s = s + s;
-            let m = xx + xx + xx + $name::curve_constant_a() * zz.square();
-            let x3 = m.square() - (s + s);
-            let a = a + a;
-            let a = a + a;
-            let a = a + a;
+            let s = ((self.x + yy).square() - xx - a).double();
+            let m = xx.double() + xx + $name::curve_constant_a() * zz.square();
+            let x3 = m.square() - s.double();
+            let a = a.double();
+            let a = a.double();
+            let a = a.double();
             let y3 = m * (s - x3) - a;
-            let z3 = (self.x + self.y).square() - yy - zz;
+            let z3 = (self.y + self.z).square() - yy - zz;
 
             let tmp = $name {
                 x: x3,
diff --git a/src/pasta/pallas.rs b/src/pasta/pallas.rs
index 4af4ae7..2bce353 100644
--- a/src/pasta/pallas.rs
+++ b/src/pasta/pallas.rs
@@ -14,6 +14,79 @@ pub type Point = Ep;
 /// A Pallas point in the affine coordinate space (or the point at infinity).
 pub type Affine = EpAffine;
 
+#[test]
+fn test_iso_map() {
+    use crate::arithmetic::Curve;
+
+    // This is a regression test (it's the same input to iso_map as for hash_to_curve
+    // with domain prefix "z.cash:test", Shake128, and input b"hello").
+    let r = super::IsoEp::new_jacobian(
+        Base::from_raw([
+            0xc37f111df5c4419e,
+            0x593c053e5e2337ad,
+            0x9c6cfc47bce1aba6,
+            0x0a881e4d556945aa,
+        ]),
+        Base::from_raw([
+            0xf234e04434502b47,
+            0x6979f7f2b0acf188,
+            0xa62eec46f662cb4e,
+            0x035e5c8a06d5cfb4,
+        ]),
+        Base::from_raw([
+            0x11ab791d4fb6f6b4,
+            0x575baa717958ef1f,
+            0x6ac4e343558dcbf3,
+            0x3af37975b0933125,
+        ]),
+    )
+    .unwrap();
+    let p =
+        super::hashtocurve::iso_map::<_, Affine, super::IsoEpAffine>(&r, &Ep::ISOGENY_CONSTANTS);
+    let (x, y, z) = p.jacobian_coordinates();
+    assert!(
+        format!("{:?}", x) == "0x318cc15f281662b3f26d0175cab97b924870c837879cac647e877be51a85e898"
+    );
+    assert!(
+        format!("{:?}", y) == "0x1e91e2fa2a5a6a5bc86ff9564ae9336084470e7119dffcb85ae8c1383a3defd7"
+    );
+    assert!(
+        format!("{:?}", z) == "0x1e049436efa754f5f189aec69c2c3a4a559eca6a12b45c3f2e4a769deeca6187"
+    );
+}
+
+#[test]
+fn test_iso_map_identity() {
+    use crate::arithmetic::Curve;
+
+    let r = super::IsoEp::new_jacobian(
+        Base::from_raw([
+            0xc37f111df5c4419e,
+            0x593c053e5e2337ad,
+            0x9c6cfc47bce1aba6,
+            0x0a881e4d556945aa,
+        ]),
+        Base::from_raw([
+            0xf234e04434502b47,
+            0x6979f7f2b0acf188,
+            0xa62eec46f662cb4e,
+            0x035e5c8a06d5cfb4,
+        ]),
+        Base::from_raw([
+            0x11ab791d4fb6f6b4,
+            0x575baa717958ef1f,
+            0x6ac4e343558dcbf3,
+            0x3af37975b0933125,
+        ]),
+    )
+    .unwrap();
+    let r = (r * -Fq::one()) + r;
+    assert!(bool::from(r.is_on_curve()));
+    let p =
+        super::hashtocurve::iso_map::<_, Affine, super::IsoEpAffine>(&r, &Ep::ISOGENY_CONSTANTS);
+    assert!(bool::from(p.is_on_curve()));
+}
+
 #[test]
 fn test_map_to_curve_pallas() {
     use crate::arithmetic::Curve;
@@ -31,4 +104,7 @@ fn test_map_to_curve_pallas() {
     assert!(
         format!("{:?}", z) == "0x1e049436efa754f5f189aec69c2c3a4a559eca6a12b45c3f2e4a769deeca6187"
     );
+    assert!(bool::from(p.is_on_curve()));
+    let p = (p * -Fq::one()) + p;
+    assert!(bool::from(p.is_on_curve()));
 }