┌────────────────────────────────────────────────────────────────────┐
│ Sage Version 6.0, Release Date: 2013-12-17                         │
│ Type "notebook()" for the browser-based notebook interface.        │
│ Type "help()" for help.                                            │
└────────────────────────────────────────────────────────────────────┘
sage: 

sage: K = QQ[(1+2^(1/3))^(1/2)]
sage: a = K.gen()
sage: K.ring_of_integers().gens()
[1, a, a^2, a^3, a^4, a^5]
sage: K.trace_pairing([1,a,a^2,a^3,a^4,a^5])
[  6   0   6   0   6   0]
[  0   6   0   6   0  18]
[  6   0   6   0  18   0]
[  0   6   0  18   0  54]
[  6   0  18   0  54   0]
[  0  18   0  54   0 126]
sage: K.trace_pairing([1,a,a^2,a^3,a^4,a^5]).det()
2239488
sage: K.discriminant()
2239488
sage: QQ[3^(1/2)].discriminant()
12
sage: QQ[5^(1/2)].discriminant()
5
sage: QQ[10^(1/3)].discriminant()
-300
sage: