# How to get a solution from an ideal in a polynomial ring when it is nonzero codimensional?

Below is an ideal Id of codimension 3 (in normal-basis form). It is not 0-codim so we cannot do `Id.variety()`

.
Note that Id.dimension() provides the dimension of the ring modulo the ideal Id.

**Question**: How can we still get a solution?

```
sage: R.<s2,s3,a0, a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11, a12, a13, a14, a15, b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, c0, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15>=PolynomialRing(QQ)
sage: Id=Ideal(b9^2*b14^2*c0 - 1/9*b6*b11^3*c1, b8*b10^3*c1 - 9*b9^2*c0*c14^2, b10^4 + 9*b9^2*c14^2, b11^4 + 9*b9^2*b14^2, b14^2*c9^2 + 1/9*c10^4, c11^4 + 9*c9^2*c14^2, b8*b10*c1*c15 + 1/2*b9*c0*c14, b6*b11*c1*c15 - 1/2*b9*b14*c0, b14*c0*c1*c15 + 1/6*b4*c8*c10, b4*c8*c10*c15 - 1/6*b14*c0*c1, c3*c8*c10*c15 + 1/2*a14*b14*c4, b4*c6*c11*c15 + 1/6*c0*c1*c14, c3*c6*c11*c15 - 1/2*a14*c4*c14, c0*c1*c14*c15 - 1/6*b4*c6*c11, b8*b11^2 - 1/2*b6*b9, b8*b11*c1 + 1/2*b9*c0, b9*c0*c1 - 1/2*a14*b4, c0*c1*c9 - b3*c8*c11, b6*c1*c10 + 3*b14*c0*c15, c0*c1*c10 - 1/2*b3*c8, b4*c10^2 + 3*b3*b14*c15, b6*c10^2 - 3*b14*c8*c15, c3*c10^2 + 3*b14*c4*c15, b8*c1*c11 - 3*c0*c14*c15, c0*c1*c11 - 1/2*b3*c6, b4*c8*c11 + 1/6*c0*c1, c3*c8*c11 - 1/2*a14*c4, b4*c11^2 - 3*b3*c14*c15, b8*c11^2 + 3*c6*c14*c15, c3*c11^2 - 3*c4*c14*c15, c8*c11^2 - 1/2*c6*c9, b10^2*c15 - 1/2*b9*c14, b11^2*c15 + 1/2*b9*b14, a14*b14*c15 + 1/18*b6*c10, b9*b14*c15 - 1/18*b11^2, b14*c9*c15 - 1/18*c10^2, b6*c10*c15 - 1/2*a14*b14, c10^2*c15 + 1/2*b14*c9, b8*c11*c15 + 1/2*a14*c14, c11^2*c15 - 1/2*c9*c14, a14*c14*c15 - 1/18*b8*c11, b9*c14*c15 + 1/18*b10^2, c9*c14*c15 + 1/18*c11^2, s2^2 - 2, s3^2 - 3, a6^2 - 1/12, a6*a8 - 1/2*a14, a8^2 - 1/12, a6*a14 - 1/6*a8, a8*a14 - 1/6*a6, a14^2 - 1/36, a6*b0 - a8*b1, a8*b0 - a6*b1, a14*b0 - 1/6*b1, b0^2 + 1/2*c3, a14*b1 - 1/6*b0, b0*b1 + 3*a14*c3, b1^2 + 1/2*c3, a14*b3 + 1/3*c0*c1, b0*b3 + 1/3*c0, b1*b3 + 1/3*c1, b0*b4 - 2*b9*c0, b1*b4 - 2*b9*c1, a6*b6 + a8*b11, a8*b6 + a6*b11, a14*b6 + 1/6*b11, b0*b6 + b1*b11, b1*b6 + b0*b11, b3*b6 + b4*c8, b6^2 - b11^2, a6*b8 + a8*b10, a8*b8 + a6*b10, a14*b8 + 1/6*b10, b0*b8 + b1*b10, b1*b8 + b0*b10, b3*b8 + b4*c6, b6*b8 - 1/2*b9, b8^2 - b10^2, a14*b9 + 1/3*b8*b11, b3*b9 + 1/6*b4, a14*b10 + 1/6*b8, b3*b10 + b4*c11, b6*b10 - b8*b11, a14*b11 + 1/6*b6, b3*b11 + b4*c10, b10*b11 - 1/2*b9, b8*b14 + b6*c15, b10*b14 + b11*c15, a6*c0 - a8*c1, a8*c0 - a6*c1, a14*c0 - 1/6*c1, b0*c0 - 1/6, b1*c0 - a14, b6*c0 + b11*c1, b8*c0 + b10*c1, b10*c0 + b8*c1, b11*c0 + b6*c1, c0^2 + 1/2*b3, a14*c1 - 1/6*c0, b0*c1 - a14, b1*c1 - 1/6, c1^2 + 1/2*b3, b3*c3 - 1/9, b4*c3 + 2/3*b9, c0*c3 + 1/3*b0, c1*c3 + 1/3*b1, b3*c4 + 2/3*c9, b4*c4 - 1/9, b6*c4 + c3*c8, b8*c4 + c3*c6, b9*c4 + 1/6*c3, b10*c4 + c3*c11, b11*c4 + c3*c10, c0*c4 - 2*b0*c9, c1*c4 - 2*b1*c9, a6*c6 + a8*c11, a8*c6 + a6*c11, a14*c6 + 1/6*c11, b0*c6 + b1*c11, b1*c6 + b0*c11, b6*c6 - 1/12, b8*c6 + 3*c14*c15, b9*c6 - 1/6*b8, b10*c6 - b8*c11, b11*c6 + 1/2*a14, b14*c6 + c8*c15, c0*c6 + c1*c11, c1*c6 + c0*c11, c6^2 - c11^2, a6*c8 + a8*c10, a8*c8 + a6*c10, a14*c8 + 1/6*c10, b0*c8 + b1*c10, b1*c8 + b0*c10, b6*c8 - 3*b14*c15, b8*c8 - 1/12, b9*c8 - 1/6*b6, b10*c8 + 1/2*a14, b11*c8 - b6*c10, c0*c8 + c1*c10, c1*c8 + c0*c10, c6*c8 - 1/2*c9, c8^2 - c10^2, a14*c9 + 1/3*c8*c11, b4*c9 + 1/6*b3, b6*c9 - 1/6*c8, b8*c9 - 1/6*c6, b9*c9 - 1/36, b10*c9 - 1/6*c11, b11*c9 - 1/6*c10, c3*c9 + 1/6*c4, a14*c10 + 1/6*c8, b8*c10 + 1/2*a14, b9*c10 - 1/6*b11, b10*c10 - 1/12, b11*c10 - 3*b14*c15, c6*c10 - c8*c11, a14*c11 + 1/6*c6, b6*c11 + 1/2*a14, b9*c11 - 1/6*b10, b10*c11 + 3*c14*c15, b11*c11 - 1/12, b14*c11 + c10*c15, c10*c11 - 1/2*c9, b6*c14 - b8*c15, b11*c14 - b10*c15, b14*c14 - 1/36, c8*c14 - c6*c15, c10*c14 - c11*c15, c15^2 + 1/36, a0 - 1/2, a1 + 1/2, a2 + 2*a14, a3 - 1/3, a4, a5 + a6, a7 + a8, a9 + 2*a14, a10 - 1/3, a11 - 1/3, a12 + 1/6, a13 - a14, a15 + 1/6, b2 + 1/3, b5, b7 - 1/6, b12 + b14, b13 + c15, b15 + c15, c2 + 1/3, c5, c7 - 1/6, c12 + c14, c13 - c15)
sage: Id.dimension()
3
```