Different number field discriminants in Sage and Magma

Question

I have been working with towers of number fields. My problem is that when computing the absolute discriminant of such number fields, Sage and Magma give different values. You can see this by doing in Sage

di = [2,3,5,7]

K.< a > = NumberField([x^2 -p for p in di])

L = K.absolute_field('b')

L.discriminant()

which gives $63456228123711897600000000$ and in Magma:

Zx< x > := PolynomialRing(Integers());

K := NumberField(x^16 - 136*x^14 + 6476*x^12 - 141912*x^10 + 1513334*x^8 - 7453176*x^6 + 13950764*x^4 - 5596840*x^2 + 46225);

Discriminant( K );

which gives 5599292204088795725124575470812804054972446039778078505738080537221656920845173082195169325791965321100902635929600000000000000000000000000.

What different types of discriminants are Sage and Magma computing? Is one of them wrong?

Answer 1

The larger number is the discriminant of the defining polynomial, while the smaller number is the discriminant of the number field. In sage:

sage: K.<a> = NumberField([x^2 -p for p in di])
sage: L.<b> = K.absolute_field()
sage: L.discriminant()
63456228123711897600000000
sage: L.polynomial().discriminant()
5599292204088795725124575470812804054972446039778078505738080537221656920845173082195169325791965321100902635929600000000000000000000000000

Answer 2

Magma does know how to compute the discriminant of a number field $K$. The command (which I found in https://people.maths.bris.ac.uk/~matyd/amsmagma/) is

Discriminant(Integers(K))

But yes, it's very bad form for Magma to return the discriminant of $P \in {\bf Z}[X]$ when asked for the discriminant of the number field ${\bf Q}[X] / (P)$.