There are enough points, but how do you determine the equation?

From the given points, we get the following equations

$36b + 6e+f=0$

$36a+6d+f=0$

$a+9b+3c+d+3e+f=0$

$9a+b+3c+3d+e+f=0$

$100a+b-10c+10d-e+f=0$

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These are $5$ equations and $6$ variables, so you can't get values of all of them exactly. But we can eliminate $5$ variables, giving each variable in terms of the $6^{th}$ variable. After getting each variable in terms of $a$ (Which is a long algebraic manipulation, too long to type), we get

$b=a , c=\dfrac{52a}{11} , d=e=\dfrac{-65a}{11} , f = \dfrac{-6}{11}$

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Now, as a given statement is $gcd(a,b,c,d,e,f)=1$ , we conclude that $a,b,c,d,e,f$ are integers, so for smallest value, we will multiply it by 11 directly (no one said it should be positive)

Giving $a=11, b=11 , c=52 , d=-65 , e =-65, f =-6$

They all sum up to give $\boxed{-62}$

Note:- Using this wolfram alpha input is easier but cheating way.

You need to specify an unique answer. As the problem is stated now, both $62$ and $-62$ are correct answers.