We give birational parametrizations of pairs
(
C
,
T
)
(C,T)
where
T
T
is a
5
5
-torsion point on the Jacobian of a genus-
2
2
curve
C
C
, possibly satisfying one or more of the following additional conditions:
T
=
(
P
)
−
(
P
0
)
T = (P) - (P_0)
for some points
P
0
,
P
∈
C
P_0,P \in C
with
P
0
P_0
Weierstrass;
C
C
has one or more additional rational Weierstrass points;
J
a
c
(
C
)
Jac(C)
has real multiplication by
(
1
+
5
)
/
2
(1+\sqrt {5})/2
, and
T
T
is a
5
\sqrt {5}
-torsion point.