0          $\begin{pmatrix}          3                 $A_2$
                                        0 & 1 \\                      
                                      -1 & -1 \\                      
                                  \end{pmatrix}$                      

     $IV^*$          0          $\begin{pmatrix}          3                 $E_6$
                                      -1 & -1 \\                      
                                        1 & 0 \\                      
                                  \end{pmatrix}$                      
  --------------------------------------------------------------------------------------

  : \[tablemonodromy in 2.1\]Monodromies, j-invariants and the corresponding types of the singularities of singular fibers. “Regular” for j-invariant of $I_0^*$ fiber means that j-invariant can take any finite value in $\C$ for $I_0^*$ fiber.

The types of singular fibers of elliptic surfaces can be determined from the vanishing orders of the coefficients of the Weierstrass equations. The correspondences of the fiber types and the vanishing orders of the Weierstrass coefficients are shown in Table \[tableWeierstrasscoefficients in 2.1\].

       Fiber Type        Ord($f$)   Ord($g$)   Ord($\Delta$)
  --------------------- ---------- ---------- ---------------
         $I_0 $          $\ge 0$    $\ge 0$          0
    $I_n $ ($n\ge 1$)       0          0            $n$
          $II $          $\ge 1$       1             2
         $III $             1       $\ge 2$          3
          $IV $          $\ge 2$       2             4
         $I_0^*$         $\ge 2$       3             6
                            2       $\ge 3$   
   $I^*_m$ ($m \ge 1$)      2          3           $m+6$
         $IV^*$          $\ge 3$       4             8
         $III^*$            3       $\ge 5$          9
         $II^*$          $\ge 4$       5            10

  : \[tableWeierstrasscoefficients in 2.1\]List of the types of the singular fibers, and the corresponding vanishing orders of the coefficients, $f,g$, of the Weierstrass equation 