ASKSAGE: Sage Q&A Forum - Latest question feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Thu, 18 Jan 2018 08:58:26 -0600Calculation of maximal order fails even when using "maximize_at_primes"https://ask.sagemath.org/question/40677/calculation-of-maximal-order-fails-even-when-using-maximize_at_primes/I would like to calculate the maximal order of a field for which I already know at which primes we should maximize.
This means that the discriminant does not have to be factored, which is normally the bottleneck of this algorithm.
I did
d = [2,3,5,7,11,13,17,19]
K.<y> = NumberField([x^2 - di for di in d], maximize_at_primes=[2])
RR = K.maximal_order()
The last command did not finish overnight, but gave the following warning:
"*** Warning: MPQS: number too big to be factored with MPQS,
giving up."
Which seems to indicate that the program is indeed trying to perform a large factorization despite the command "maximize_at_primes=[2]"
Meanwhile, Magma has no problem performing this computation in a few hours:
R<x> := PolynomialRing(Integers());
K := NumberField([x^2 - 2,x^2 - 3 , x^2-5,x^2-7,x^2 - 11,x^2 - 13 , x^2 - 17 , x^2 - 19]:Abs);
O := MaximalOrder(K: Ramification := [2]);
Am I doing something wrong ?
JF BiassebiasseThu, 18 Jan 2018 08:58:26 -0600https://ask.sagemath.org/question/40677/Is there any way to find decomposition group and ramification groupshttps://ask.sagemath.org/question/35472/is-there-any-way-to-find-decomposition-group-and-ramification-groups/Let $L/K$ be a Galois extension of number fields with Galois group $G$. Let $O_K$ and $O_L$ be the ring of algebraic integers of $K$ and $L$ respectively. Let $P\subseteq O_K$ be a prime. Let $Q\subseteq O_L$ be a prime lying over $P$.
The decomposition group is defined as $$D(Q|P)=\lbrace \sigma\in G\text{ }|\text{ }\sigma(Q)=Q\rbrace$$
The $n$-th ramification group is defined as $$E_n(Q|P)=\lbrace \sigma\in G:\sigma(a)\equiv a\text{ mod } Q^{n+1}\text{ for all } a\in O_L\rbrace$$
I want to compute the decomposition group and ramification groups of the cyclotomic field $\mathbb{Q}(\zeta)$ over $\mathbb{Q}$ where $\zeta$ is a root of unity.
How to do this ? Any idea ?nebuckandazzerMon, 07 Nov 2016 09:50:42 -0600https://ask.sagemath.org/question/35472/