Let $X$ be an elliptic K3 surface with $D\_{14}$ singular fiber. Do you know an explicit equation for such $X$? Also, how many disjoint sections such fibration admits? Any reference would be greatly appreciated. 



K3 surface with $D_{14}$ singular fiber
[In comments **guest2014** amended the question to ask not for a
$D\_{14}$ fiber but for $I^\*\_{14}$, a.k.a. $\tilde D\_{18}$]


The elliptic surface 
$$
X : y^2 = x^3 + (t^3+2t) x^2 - 2(t^2+1)x + t
$$
over ${\bf C}(t)$ has a $I^\*\_{14}$ fiber at $t=\infty$.
(Note that the right-hand side is a cubic in $x$ whose discriminant
$4t^4 + 13 t^2 + 32$ has degree only $4$ in $t$, where a generic
K3 surface would have discriminant of degree $24$.)
$X$ is the unique such surface with a section, up to isomorphism.
The group of sections is trivial: the zero-section and the components of
the $\tilde D\_{18}$ fiber already generate a lattice 
${\rm II}\_{1,1} \oplus D\_{18} \langle -1 \rangle$ of rank $20$ in
${\rm NS}(X)$, so the Mordell-Weil rank is zero (else the Picard number
would exceed $20$), and if there were
a $2$-torsion section then the Neron-Severi lattice would be
${\rm II}\_{1,1} \oplus L \langle -1 \rangle$ for some
even unimodular lattice $L$ of rank $18$, which is impossible because
$18 \not\equiv 0 \bmod 8$.


[*added a bit later*: For the situation in positive characteristic see
[Schütt's paper](http://arxiv.org/abs/math/0509540),
which cites Shioda's note



> 
> Shioda, T.: The elliptic K3 surfaces with a maximal singular fibre, 
>  *C. R. Acad. Sci. Paris, Ser. I*, **337** (2003), 461-466.
> 
> 
> 


for the description of elliptic surfaces with a maximal singular fiber
in characteristic zero.]


