K3 surface with $D_{14}$ singular fiber

Question

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.

Answer 1

[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, 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.]