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The degree of an ideal I in a ring S is the degree of the module S/I. See degree(Module) for more details.
i1 : S = QQ[a..f];
i2 : I = ideal(a^5, b^5, c^5, d^5, e^5);

o2 : Ideal of S
i3 : degree I

o3 = 3125
i4 : degree(S^1/I)

o4 = 3125
If the ideal is not homogeneous, then the degree returned is the degree of the ideal of initial monomials (which is homogeneous). If the monomial order is a degree order (the default), this is the same as the degree of the projective closure of the zero set of I.
i5 : I = intersect(ideal(a-1,b-1,c-1),ideal(a-2,b-1,c+1),ideal(a-4,b+7,c-3/4));

o5 : Ideal of S
i6 : degree I

o6 = 3

See also

Ways to use this method: