Least number $k>11$, such that $k+n$ is prime for $n=0,2,6,8,12,18,20,26,30,32$ The least integer $k>11$, for which $k+n$ is prime for $n=0,2,6,8,12,18,20,26,30,32$, is, according to my search, $k=33,081,664,151$.
The numbers form a prime constellation with length $10$ and width $32$.

$1)$ Is my number the smallest one ?
$2)$ How can the smallest number be found without brute force ?

Looking at the residue classes $2$,$3$,$5$ and $7$, I found out that $k$
must be of the form $210k+11$, but I did not get further restrictions
for $k$. Of course, I could continue with the residue classes modulo $11,13,... $. Is this the best and only way to restrict $k$, or are there more powerful
methods ?

You say you got nothing beyond k = 210n + 11, but we also know that k modulo 11 is 6 or 8, which makes k = 2310n + 1271 or 2310n + 1691, and there are only 5 values modulo 13, so there are 10 values modulo 30,030. Plus 7 values modulo 17, so there are 70 values modulo 510,510, or one in 7293. 
