In this paper, we find bases for the Mordell-Weil groups of a family of elliptic surfaces. In particular, let
E
(
a
,
b
)
→
B
{E_{(a,b)}} \to B
be the elliptic surface given by
\[
y
2
=
4
[
x
3
−
∑
i
=
0
2
a
i
u
i
x
+
∑
j
=
0
3
b
j
u
j
]
.
{y^2} = 4\left [ {{x^3} - \sum \limits _{i = 0}^2 {{a_i}{u^i}x + } \sum \limits _{j = 0}^3 {{b_j}{u^j}} } \right ].
\]
If the elliptic surface has Mordell-Weil rank 4 over
C
{\mathbf {C}}
, then we find a basis
{
σ
i
=
(
x
i
,
y
i
)
|
1
≤
i
≤
4
}
\{ {\sigma _i} = ({x_i},{y_i})|1 \leq i \leq 4\}
with
x
i
{x_i}
and
y
i
{y_i}
, linear in
u
u
. We do this by finding a parametrization of this family of elliptic surfaces; furthermore, if the parameters are rational numbers, then the Mordell-Weil group is rational over
Q
{\bf {Q}}