SymbolicData

Sage has many optional packages, about 50 actually. However, it is not necessarily well known what some of these do. For example, recently William asked me to fix some issue in the optional database_symbolic_data (still needing review btw. hint hint) package which bought it back to my attention. Shortly after, Burcin mentioned to me that he spent considerable time copy’n’pasting some standard ideals from some package to Sage’s input format. Time which he probably wouldn’t have spent if he’d known about the SymbolicData.org database and it’s Sage interface. Here’s in it action:

sage: install_package('database_symbolic_data-20070206')
sage: sd = SymbolicData()
sage: sd.Cyclic_4
Ideal (w + x + y + z, w*x + x*y + w*z + y*z, w*x*y + w*x*z + w*y*z + x*y*z, w*x*y*z - 1) of Multivariate Polynomial Ring in w, x, y, z over Rational Field

And another example:

sage: sd.ZeroDim__example_20
Ideal (4*t2^2*t3^2 + 2*t2^2 + 5*t2*t3 + 3*t3^2 + 1, 4*t1^2*t3^2 + 3*t1^2 + 5*t1*t3 + 2*t3^2 + 1, 4*t1^2*t2^2 + 2*t1^2 + 5*t1*t2 + 3*t2^2 + 1) of Multivariate Polynomial Ring in t1, t2, t3 over Rational Field

The database contains 372 ideals. But what if you don’t like the term order or base field? You can either use

sage: I = sd.Cyclic_4
sage: I.change_ring(I.ring().change_ring(base_ring=GF(127),order="lex"))
Ideal (w + x + y + z, w*x + w*z + x*y + y*z, w*x*y + w*x*z + w*y*z + x*y*z, w*x*y*z - 1) of Multivariate Polynomial Ring in w, x, y, z over Finite Field of size 127

Or use the function get_ideal:

sage: sd.get_ideal("Cyclic_4",base_ring=GF(127),term_order="lex")
Ideal (w + x + y + z, w*x + x*y + w*z + y*z, w*x*y + w*x*z + w*y*z + x*y*z, w*x*y*z - 1) of Multivariate Polynomial Ring in w, x, y, z over Finite Field of size 127

Finally, since Sage has interfaces to many other systems this also means that it converts from SymbolicData format to these formats as well:

sage: I = sd.Cyclic_4
sage: magma(I)
Ideal of Polynomial ring of rank 4 over Rational Field
Order: Graded Reverse Lexicographical
Variables: w, x, y, z
Basis:
[
w + x + y + z,
w*x + x*y + w*z + y*z,
w*x*y + w*x*z + w*y*z + x*y*z,
w*x*y*z - 1
]