# Compute Groebner Basis of an ideal that includes parameters

I'm trying to compute the Groebner Basis of the ideal $$I=\langle f_1,f_2,f_3,f_4,f_5 \rangle$$ in $\mathbb{R}[x_1,y_1,x_2,y_2,x_3,y_3]$, where

$$ f_1=13(x_1+x_1x_2-y_1y_2+(x_1x_2-y_1 y_2)x_3-(y_1 x_2+y_2x_1)y_3)-x_0 $$ $$ f_2=13(y_1+y_1x_2+y_2x_1+(y_1 x_2+y_2x_1)x_3+(x_1x_2-y_1 y_2)y_3)-y_0$$ $$f_3=x_1^2+y_1^2-1$$ $$f_4=x_2^2+y_2^2-1$$ $$f_5=x_3^2+y_3^2-1.$$

$x_1,y_1,x_2,y_2,x_3,y_3$ are real variables and $x_0,y_0$ are some constants. I'm using lex order with $x_1>y_1>x_2>y_2>x_3>y_3$. What I've tried was

```
sage: R.<x1,y1,x2,y2,x3,y3> = PolynomialRing(QQ, order='lex')
sage: x0,y0 = var('x0,y0')
sage: f1 = 13*(x1+x1*x2-y1*y2+(x1*x2-y1*y2)*x3-(y1*x2+y2*x1)*y3)-x0
sage: f2 = 13*(y1+y1*x2+y2*x1+(y1*x2+y2*x1)*x3+(x1*x2-y1*y2)*y3)-y0
sage: f3 = x1^2+y1^2-1
sage: f4 = x2^2+y2^2-1
sage: f5 = x3^2+y3^2-1
sage: ideal(f1,f2,f3,f4,f5).groebner_basis()
```

but the output was

```
---------------------------------------------------------------------------
AttributeError Traceback (most recent call last)
<ipython-input-8-e03b88a807f0> in <module>()
----> 1 ideal(f1,f2,f3,f4,f5).groebner_basis()
sage/structure/element.pyx in sage.structure.element.Element.__getattr__ (/usr/lib/sagemath//src/build/cythonized/sage/structure/element.c:4675)()
sage/structure/misc.pyx in sage.structure.misc.getattr_from_other_class (/usr/lib/sagemath//src/build/cythonized/sage/structure/misc.c:1771)()
AttributeError: 'Ideal_generic' object has no attribute 'groebner_basis'
```

What can I do?

@ilgk: in general, only tick the "community wiki" box for "meta" questions (questions about the design of the Ask Sage website).