mathematics – Ten roots challenge

Find integers $u_0,u_1,u_2,u_3,\dots,u_{10}$ such that

• $(u_{10})x^{10}+(u_9)x^9+(u_8)x^8+\dots+(u_1)x+(u_0)=0$ has ten distinct real roots, and
• $S=|u_0|+|u_1|+|u_2|+|u_3|+\dots+|u_{10}|$ is minimized

For example:

$$x^{10}-2x^9-5x^8+10x^7+7x^6-15x^5-2x^4+7x^3+0x^2-x+0=0$$

has ten distinct real roots, and $S=50$.

Can you beat $50$?

How low can you go?

p.s. This if far more subtle than a textbook style math problem.