What is the smallest $20$ digit number with at least $13$ distinct prime factors?

At least one can find some upper bound, i.e., some integer $n\ge 10^{19}$, by multiplying $13$ different primes to an integer $m$, such that $m<10^{19}$, and then using the minimal factor to obtain a number $>10^{19}$. For example, $$ m=2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13\cdot 17\cdot 19\cdot 23\cdot 29\cdot 31\cdot 37\cdot 43=319091739796830 $$ gives $$ n=31339m=10000016033492855370, $$ having again exactly $13$ different prime divisors. Varying the primes of $m$, one can have better bounds.

I got $$10\,000\,000\,000\,255\,252\,260=2^2\cdot 3\cdot5\cdot 7\cdot13\cdot19\cdot37\cdot43\cdot61\cdot73\cdot101\cdot107\cdot1259$$ (with $10$ zeros after the initial $1$!). This number was obtained by multiplying the primes up to $7$, resulting in $210$, and then factoring the multiples of $210$ larger than $10^{19}$ until one was found having $\geq13$ prime factors.