diff --git a/field/src/goldilocks_field.rs b/field/src/goldilocks_field.rs
index e10d0c74..4432d9c4 100644
--- a/field/src/goldilocks_field.rs
+++ b/field/src/goldilocks_field.rs
@@ -118,22 +118,6 @@ impl Field for GoldilocksField {
         reduce128(n)
     }
 
-    #[inline]
-    fn multiply_accumulate(&self, x: Self, y: Self) -> Self {
-        // u64 + u64 * u64 cannot overflow.
-        reduce128((self.0 as u128) + (x.0 as u128) * (y.0 as u128))
-    }
-}
-
-impl PrimeField for GoldilocksField {
-    fn to_canonical_biguint(&self) -> BigUint {
-        self.to_canonical_u64().into()
-    }
-}
-
-impl Field64 for GoldilocksField {
-    const ORDER: u64 = 0xFFFFFFFF00000001;
-
     #[inline]
     fn from_noncanonical_u64(n: u64) -> Self {
         Self(n)
@@ -151,6 +135,22 @@ impl Field64 for GoldilocksField {
         })
     }
 
+    #[inline]
+    fn multiply_accumulate(&self, x: Self, y: Self) -> Self {
+        // u64 + u64 * u64 cannot overflow.
+        reduce128((self.0 as u128) + (x.0 as u128) * (y.0 as u128))
+    }
+}
+
+impl PrimeField for GoldilocksField {
+    fn to_canonical_biguint(&self) -> BigUint {
+        self.to_canonical_u64().into()
+    }
+}
+
+impl Field64 for GoldilocksField {
+    const ORDER: u64 = 0xFFFFFFFF00000001;
+
     #[inline]
     unsafe fn add_canonical_u64(&self, rhs: u64) -> Self {
         let (res_wrapped, carry) = self.0.overflowing_add(rhs);
diff --git a/field/src/types.rs b/field/src/types.rs
index 0ae31847..2c0d15c7 100644
--- a/field/src/types.rs
+++ b/field/src/types.rs
@@ -341,6 +341,26 @@ pub trait Field:
     /// Returns `n % Self::characteristic()`.
     fn from_noncanonical_u128(n: u128) -> Self;
 
+    /// Returns `x % Self::CHARACTERISTIC`.
+    ///
+    /// Implemented by default via `from_noncanonical_u128`
+    /// Override, if you have a faster implementation for your field.
+    fn from_noncanonical_u64(n: u64) -> Self {
+        Self::from_noncanonical_u128(n as u128)
+    }
+
+    /// Returns `n` as an element of this field.
+    ///
+    /// Implemented by default via `from_noncanonical_u128` and case analysis.
+    /// Override, if you have a faster implementation for your field.
+    fn from_noncanonical_i64(n: i64) -> Self {
+        if n < 0 {
+            -Self::from_noncanonical_u128((n as i128).unsigned_abs())
+        } else {
+            Self::from_noncanonical_u128(n as u128)
+        }
+    }
+
     /// Returns `n % Self::characteristic()`. May be cheaper than from_noncanonical_u128 when we know
     /// that `n < 2 ** 96`.
     #[inline]
@@ -501,14 +521,6 @@ pub trait PrimeField: Field {
 pub trait Field64: Field {
     const ORDER: u64;
 
-    /// Returns `x % Self::CHARACTERISTIC`.
-    // TODO: Move to `Field`.
-    fn from_noncanonical_u64(n: u64) -> Self;
-
-    /// Returns `n` as an element of this field.
-    // TODO: Move to `Field`.
-    fn from_noncanonical_i64(n: i64) -> Self;
-
     /// Returns `n` as an element of this field. Assumes that `0 <= n < Self::ORDER`.
     // TODO: Move to `Field`.
     // TODO: Should probably be unsafe.