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Hello, I am trying to find all rational periodic points of a polynomial. To specify: a periodic point is the point that satisfy $f^n(x)=x$. It is related to dynamical systems in fact. So the current codes that I used are following:

A.<z> = AffineSpace(QQ, 1)
f = DynamicalSystem_affine([2*z^3-3*z^2+1/2])
x=f.dynatomic_polynomial(2)
x.factor()

With this I can find its dynatomic polynomial and factorize it and find rational roots of this polynomial. So this roots corresponds to periodic point of the polynomial of given period. In particular dynatomic polynomial is the polynomial of the form $$\phi_{n,f}(x)=\prod_{d|n}(f^d(x)-x)^{\mu(n/d)}$$ n is period, f is your polynomial and $\mu$ is the mobius function.

But with this code I can find periods up to 8 because of memory limit. The other code that I used is

R.<x> = QQ[]
K.<i> = NumberField(xˆ2+1)
A.<z> = AffineSpace(K,1)
f = DynamicalSystem([zˆ2+i], domain=A)
f.orbit(A(0),4)

But in fact it doesn't fit my purposes.

I have codes that I can get limited information. For example checking up to a period is not advisable. If you know a little bit arithmetic dynamics, you can see what I mean. Silverman-Morton conjecture plays an important role here.

I am waiting for your answers. Thank you so much.

I cannot post the polynomial here because it is too long. But you can compute with wolfram, sage or magma. Let f(x)=ax^3+bx+c be a cubic polynomial with rational coefficients. Dynatomic polynomial of f(x) is (f^3(x)-x)/(f(x)-x). Here f^3(x) is the composition of f 3 times. I am searching galois group of this dynatomic polynomial.

Here is the code https://github.com/davidkrumm/local-global/blob/master/Failure%20of%20local-global%20principle (https://github.com/davidkrumm/local-g...)

I run this code for galois_candidates(3,3) First 3 corresponds to the period, second corresponds to the degree of f(x). As a sample you can run it for (1,3), (2,3) or for quadratic polynomials (1,2) up to (5,2). If you run for quadratics you can see what I want to compute. Thanks for your help.

Hello everyone, I am using a Sage code to compute galois group of a polynomial of degree 24. It is a symbolic polynomial so this code gives possible galois groups. When I run it for lower degrees it gives output but when the polynomial has degree 24 it doesn't compute. Because it computes subgroups of a group of order over 264 million. I know it is very hard to find these groups but i need it. I tried so hard for 3 weeks or more. I use sagemathcell, cocalc and Oracle Virtual Box. I tried so many things but they didn't work. It gives GAPerror: reached pre-set memory limit OR runtime error Gap produced error output, error, variable $sage37 must have a value. I guess the problem is computing conjugacy classes of subgroups. How can I eliminate this Gap error?

P.S1: I am very beginner at Sage. My knowledge about coding, programming etc. is very small. P.S2: I cannot write the code here because it doesn't belong to me. I found it on internet. But if you want to see I can send you the link.