On Thu, Oct 3, 2013 at 12:15 PM, Suman Ahmed <[email protected]> wrote:
> Respected Prof. John Cremona,
>
> Thanks for your reply. But please explain me the following -
>
> 1) Why the roots are given as a 2-tuple and why 1 is appearing always as
> the second coordinate ? What does 1 signify here ?
It is the multiplicity of the roots.
> 2) What does " *I " mean in " sage: f.roots(CC) " ?
I=sqrt(-1)
Regards,
(hopefully) Respected Prof. William Stein
>
> Regards,
>
> Suman
>
>
>
> On Thursday, October 3, 2013 2:23:58 PM UTC+5:30, John Cremona wrote:
>>
>> There does not seem to be a function returning the splitting field of
>> a reducible polynomial. For irreducible f you can do this:
>>
>> sage: Zx.<x> = ZZ[]
>> sage: f = x^3-2
>> sage: K.<a> = f.root_field()
>> sage: K
>> Number Field in a with defining polynomial x^3 - 2
>> sage: L.<b> = K.galois_closure(); L
>> Number Field in b with defining polynomial x^6 + 40*x^3 + 1372
>> # alternative: L = K.galois_group(names='b').splitting_field()
>> sage: f.roots(K)
>> [(a, 1)]
>> sage: f.roots(L)
>> [(1/84*b^4 + 13/42*b, 1), (-1/252*b^4 - 55/126*b, 1), (-1/126*b^4 +
>> 8/63*b, 1)]
>>
>> From your question it looks is if you want the 2-division field of an
>> elliptic curve. In that case you can get the polynomial this way:
>>
>> sage: E = EllipticCurve([0,0,0,0,-2]); E
>> Elliptic Curve defined by y^2 = x^3 - 2 over Rational Field
>> sage: f = E.division_polynomial(2); f
>> 4*x^3 - 8
>> sage: f = f.monic(); f
>> x^3 - 2
>>
>> For any univariate polynomial f, f.roots() gives its roots over the
>> base field and f.roots(K) the roots over an extension K of the base.
>> Here, f has not rational roots, 1 root in K, 3 in L, etc:
>>
>> sage: f.roots()
>> []
>> sage: f.roots(K)
>> [(a, 1)]
>> sage: f.roots(L)
>> [(1/84*b^4 + 13/42*b, 1), (-1/252*b^4 - 55/126*b, 1), (-1/126*b^4 +
>> 8/63*b, 1)]
>> sage: f.roots(RR)
>> [(1.25992104989487, 1)]
>> sage: f.roots(CC)
>> [(1.25992104989487, 1),
>> (-0.629960524947437 - 1.09112363597172*I, 1),
>> (-0.629960524947437 + 1.09112363597172*I, 1)]
>>
>> John Cremona
>>
>> On 2 October 2013 12:22, Suman Ahmed <[email protected]> wrote:
>> > I need to calculate the splitting field of a polynomial over Z ( in my
>> > case, the polynomial is the Weierstrass equation of an elliptic curve ),
>> > the
>> > degree of extension of the splitting field over Q and the roots ( if it's
>> > not possible to find out all of the roots, at least one can find one of the
>> > roots ). So I will request to kindly inform me about how to compute them on
>> > SAGE & whether I can perform the calculations on the online SAGE notebook ?
>> >
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--
William Stein
Professor of Mathematics
University of Washington
http://wstein.org
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