Questions tagged [sage]
Sage is a mathematical software system, and this tag is intended for questions involving this software in a substantive way. This tag should hardly ever be the only tag of a question; typically there should be additional tags to indicate the mathematical content of the question. Please note that questions that are purely support-questions on Sage are not a good fit for this site.
56 questions
3
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Compute generators for group of totally positive units of a number field?
Given a number field $K$, I would like to compute (in Sage) generators for the group of totally positive units of $K$.
Update: I've tried some code (details below), which I've received some help on in ...
- 111
1
vote
0
answers
90
views
Can computer algebra system compute Galois group/splitting field of a polynomial over $p$-adic number field of higher degree?
I am looking for a computer algebra system that checks if the splitting fields of two polynomials over a $p$-number field are the same or not. At least, knowing their splitting fields are isomorphic ...
- 195
2
votes
1
answer
134
views
On a efficient algorithm for factoring bivariate polynomials modulo composite modulus assuming the solution is unique
We found and implemented in sage efficient algorithm for factoring
bivariate polynomials modulo composite modulus assuming the solution is unique up to a constant factor.
More formally let $K=\mathbb{...
- 25.4k
4
votes
0
answers
127
views
Minimal Model for $\mathbb{CP}^2 \# \mathbb{CP}^2 \# \mathbb{CP}^2$
I'm a grad student studying non-negative curvature on simply-connected manifolds and the conjectured relationship with rational homotopy theory.
The rational homotopy groups (and so the number of ...
- 183
1
vote
1
answer
118
views
Re: Vertices of hyperbolic triangle with given angles [closed]
I'm working on visualizing hyperbolic triangles given angles following a previous discussion on MathOverflow. The algorithm, as outlined in this MathOverflow answer, involves computing the side ...
- 129
1
vote
1
answer
141
views
Factorial series $j(D)=\sum_{n=1}^\infty \frac{1}{(n^D)!}$ and hypergeometric functions
For positive integer $D$, define $j(D)=\sum_{n=1}^\infty \frac{1}{(n^D)!}$.
For $D \le 6$, sage finds closed form in terms of hypergeometric functions
at algrebraic arguments and fails to find closed ...
- 25.4k
0
votes
0
answers
112
views
Software for Intersection of Ideals in Noncommutative Polynomial algebra
I am looking for software which can compute an intersection of ideals (in particular right ideals) in a noncommutative polynomial algebra and then find its Gröbner Basis.
Most software somehow does ...
1
vote
0
answers
133
views
Hall-Littlewood polynomials with sage
I need to make some computations with Sage involving Hall-Littlewood polynomials. I couldn't find any satisfying information in the Sage manuals/tutorials that I found on the internet. What I found is ...
- 6,349
19
votes
1
answer
711
views
Discrepancy in Magma's calculation and Sage's of elliptic curve?
$\DeclareMathOperator\Sha{Sha}$I calculated the Tate–Shafarevich group $\Sha(E/K)[2]$ of the elliptic curve $E:y^2=x^3+17x$ over $K=\Bbb{Q}(\sqrt{-37})$.
I calculated that by hand and I reached the ...
- 1,541
3
votes
1
answer
200
views
Computations of half-integer forms in SAGE/Magma
I am currently going through Shimura's paper on half-integer weight modular forms. I would like to understand given a 𝑞-expansion of half-integral weight modular forms of arbitrary level and ...
- 63
1
vote
0
answers
204
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Units in group rings in SAGE
Is there a recorded/known SAGE code to compute units in integral group rings for finite abelian groups ?
I would be happy with a code that only works for cyclic groups. I sort of know how to ...
- 15.9k
3
votes
0
answers
152
views
Disconnected reductive algebraic groups in Sage
All simply connected split simple groups have been implement on Sage and it is possible to find their highest roots, fundamental weights, Dynkin diagrams or compute the tensor of two of their ...
- 370
0
votes
1
answer
663
views
How does Sage order the elements of the symmetric group?
In Sage, the symmetric group is a list. For instance if G = SymmetricGroup(3), we have
\begin{align*}
G[0] & = e \\
G[1] & = (1,3,2)\\
G[2] & = (1,2,3) \\
G[3] &= (2,3)\\
G[4] &= (...
- 197
4
votes
0
answers
193
views
Is it possible to compute Lie bialgebra structures with SageMath?
Is it possible to use SageMath (or some Linux open source program) to compute the bialgebra structures on a given finite dimensional Lie algebra? I wonder if such program can compute all the ...
- 523
10
votes
1
answer
319
views
Construction of skew-Hadamard matrix of order 292
I am currently looking into how to construct a skew-Hadamard matrix of order 292. Where can I find such construction?
According to multiple papers (e.g. Koukouvinos and Stylianou - On skew-Hadamard ...
- 261
8
votes
2
answers
754
views
How to get the dimension of Atkin-Lehner eigenspace or do you have any data already obtained?
I need the dimensions of the Atkin-Lehner eigenspace for the paper I'm writing.
As is well known, the cuspidal space $S_{k}(\Gamma_{0}(N))$ can be decomposed by Atkin Lehner involution. For example, ...
- 83
9
votes
3
answers
546
views
Why mpmath computes $\sum_{n=2}^\infty (-1)^n\log(n)=\log\left(\frac1 2 \sqrt{2} \sqrt{\pi}\right)$
Working with precision 500 decimal digits, mpmath in sage computes:
$$\sum_{n=2}^\infty (-1)^n\log(n)=\log\left(\frac1 2 \sqrt{2} \sqrt{\pi}\right).\tag{1}\label{1}$$
We believe the LHS of \eqref{1} ...
- 25.4k
13
votes
3
answers
4k
views
Is there any fast implementation of four color theorem in Python?
I'm now using scipy.spatial.Voronoi to generate a Voronoi graph, as shown here: voronoi graph. I'd like to apply the four color theorem on it, so that no adjcent regions share the same color. I ...
- 139
1
vote
0
answers
173
views
Calculating multiplication in a finite dimensional algebra over $\mathbb{Q}$
Suppose $ L $ be an extension over $ \mathbb{Q} $ of degree $ n $. Let $\{e_{1},e_{2},\dots,e_{n}\} $ be a basis of this extension. Now I know the product $ e_{i}^{2} $ and $ e_{i}e_{j} $ . So we can ...
- 923
2
votes
0
answers
313
views
Degree $8$ cyclic extension over $\mathbb{Q}$
Actually I am interested in degree $ 8 $ cyclic extension over $ \mathbb{Q} $. Let $ L $ be such extension. At first I was thinking to take basis as normal basis, as we can determine the galois group ...
- 923
1
vote
0
answers
136
views
Can PARI compute class numbers without factoring the discriminant?
When calculating properties of algebraic number fields, one of the hardest steps is factorizing the discriminant of a defining polynomial. This is necessary in the Pohst-Zassenhaus algorithm for ...
- 125
4
votes
1
answer
322
views
How to calculate genus number of number field using sage?
I am looking to find real quadratic fields whose Hilbert class field is abelian over $\Bbb Q$. Then I learned about genus numbers and genus field of the number field. It is enough to find a number ...
- 743
3
votes
1
answer
356
views
Understanding the implementation of the $p$-adic(?) sigma function in SageMath
I'm trying to understand the (pretty undocumented) method .sigma() method for formal groups of elliptic curves, as listed here. The source code looks like this:
<...
- 2,054
2
votes
1
answer
196
views
Software for $S$-unit equation
Is there any implementation available of an algorithm which solves in full generality the $S$-unit equation $x+y=1$ in a number field? It seems that Magma solves $ax+by=c$ but only in the algebraic ...
- 509
4
votes
1
answer
486
views
All rational periodic points
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 ...
- 41
3
votes
2
answers
380
views
How to find a solution of a large system of linear diophantine inequalities?
I need to find a solution (all solutions, or at least upper and lower bounds) in positive integer numbers to the system $Ax \ge f$, where $A$ is an integer matrix.
With SageMath, I solved it with the ...
4
votes
0
answers
260
views
Indexed character tables for wreath products in Sage and GAP
I am trying to obtain character table for the Hyperoctahedral group $\mathcal{H}_n$ in Sage using GAP.
This group arises as the wreath product $\mathcal{C}_2 \wr \mathcal{S}_n$, so of course I can ...
- 41
0
votes
1
answer
359
views
Mistake in SageMathCell code, finding integral points on elliptic curves [closed]
I've the following number:
$$12\left(n-2\right)^2x^3+36\left(n-2\right)x^2-12\left(n-5\right)\left(n-2\right)x+9\left(n-4\right)^2\tag1$$
Now I know that $n\in\mathbb{N}^+$ and $n\ge3$ (and $n$ has ...
- 111
5
votes
1
answer
413
views
Implementing zeta functions of algebraic varieties in SAGE
I am fairly new to sage, I was studying zeta functions of hypersurfaces over finite fields and I don't know how to compute them in Sage. Are there any packages that do most of the work, or maybe some ...
- 461
4
votes
1
answer
202
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Branching to Levi subgroups in SAGE and the circle action
In the SAGE computer package, there useful exist tools for branching representations of a simple Lie group to a Levi subgroup:
http://doc.sagemath.org/html/en/reference/combinat/sage/combinat/...
- 835
4
votes
2
answers
423
views
GAP versus SageMath for branching to Lie subgroups
Which computer package is better, GAP or SageMath, for
decomposing an irreducible representation of a (simple) Lie group
$G$ into representations of a Lie subgroup. I am most interested when
...
- 835
12
votes
0
answers
1k
views
Euler's totient function and Riemann hypothesis
I am looking for an upper-bound of the Euler's totient function $\varphi$ which would be equivalent to the Riemann hypothesis (RH). There is the following Nicolas' criterion about primorial numbers $...
1
vote
1
answer
266
views
Sage: Evaluation precision for elliptic curves over p-adic fields
Consider the elliptic curve $E: y^2 = x^3 + 23x+11$ over p-adic fields. In Sage I use:
k = GF(257)
E = EllipticCurve(k,[23,11])
kp = Qp(257,5) # 257-adic Field with capped relative ...
- 11
8
votes
3
answers
781
views
Computer program for counting graph homomorphisms
I would like to ask is there a computer program for counting graph homomorphisms?
- 4,826
3
votes
0
answers
698
views
Puzzle in 3D grid with black and white boxes, related to shelling
Consider a $n$ by $n$ by $n$ grid represented by the set of $3$-uples $S=\{1,2,\dots, n\}^3$.
A line (resp. slice) of $S$ is a subset of cardinal $n$ (resp. $n^2$) where two components (resp. one ...
25
votes
7
answers
2k
views
Number of collinear ways to fill a grid
A way to fill a finite grid (one box after the other) is called collinear if every newly filled box (the first excepted) is vertically or horizontally collinear with a previously filled box. See the ...
5
votes
0
answers
185
views
normal form for some finite groups, extending the small groups library
I am in need of a normal (that is, canonical) form for (some) finite groups, computable with - for example - gap or sage or any other freely available package. The goal is to make finite groups ...
- 5,822
11
votes
1
answer
635
views
How do computer algebra packages like Sagemath implement rank of a matrix
I am not sure if this is the right place to ask this question, but I believe there will be people here who do computations on computer algebra packages like Sage in their work.
I have been using ...
- 263
2
votes
1
answer
286
views
Memory usage of Gröbner basis computation
I've been calculating some Gröbner bases in preparation for finding non-commutative Hilbert series (and, once I recreate that, characters of group actions). Specifically, I've been using the ...
4
votes
1
answer
381
views
Existence of a non-Eulerian atomistic lattice with this property on the Möbius function
Let $L$ be a finite lattice with least element $\hat{0}$, greatest element $\hat{1}$, and Möbius function $\mu$.
Question 1: What class of lattices the following property characterizes? $$\mu(\hat{0},...
9
votes
6
answers
4k
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Computation of a minimal polynomial
It is relatively easy (but sometimes quite cumbersome) to compute the minimal polynomial of an algebraic number $\alpha$ when $\alpha$ is expressible in radicals. For example, the simple query
"...
- 1,625
5
votes
2
answers
392
views
Symmetry-finding with SAGE?
On pp. 152-3 of Hydon's Symmetry Methods for Differential Equations (2000 ed.), he lists some computer packages for symmetry-finding. This related Mathematica StackExchange question mentions the SYM ...
- 185
1
vote
1
answer
295
views
How to return elements of a given length in a symmetric group using Sage?
Let $S_n$ be the symmetric group over $\{1,2,\ldots,n\}$. How to return elements of length $m$ in $S_n$ using Sage? I try to find such function in Sage but didn't find one. Thank you very much.
Edit: ...
- 6,211
2
votes
0
answers
97
views
Is the bounded coset poset of a boolean interval of finite groups, Cohen-Macaulay?
Let $[H,G]$ be a boolean interval of finite groups and let $\hat{C}(H,G)$ be its bounded coset poset (i.e. the poset of cosets $Kg$ with $K \in [H,G]$, bounded below by $\emptyset$ and bounded above ...
3
votes
0
answers
247
views
Is there an integral simple fusion ring of multiplicity one and Frobenius type? (obvious excepted)
To avoid any confusion, we rewrite the basic definitions for a fusion ring (already written in this post).
A fusion ring is a finite dimensional complex space
$\mathbb{C}\mathcal{B}$ together ...
6
votes
0
answers
259
views
Is there an integral simple fusion ring rank<6, FPdim>60 and Frobenius type?
A fusion ring is a finite dimensional $\mathbb{Z}$-module
$\mathbb{Z}\mathcal{B}$ together with a distinguished basis
$\mathcal{B} = \{ h_1,...,h_r\}$ and fusion rules $ h_i \cdot h_j =
\sum_k n_{ij}^...
2
votes
0
answers
349
views
Basis for a set of polynomials in Sage? [closed]
I have a large set of polynomials in the coordinates $x,y,z$ in Sage, (e.g. $x^5y-3x^2y^2+2xy^3+x^2yz-y^2z$). I want to know, for example, if $x^5y$ is in the span of my set. Is there a Sage command ...
- 131
1
vote
1
answer
819
views
magma generators for unit group/ sage totally positive
Does anyone know how to find explicit generators for the unit group of a number field on magma?
For example, in sage one could do
K. = NumberField(x^3+x^2-2*x-1)
UnitGroup(K).gens()
and it ...
2
votes
2
answers
411
views
Computer algebra system that test zero divisors in a quotient algebra
I have an algebra $A$ over a Noetherian ring and an ideal $I=(x,y)$, where $x,y \in A$. I need to examine whether a polynomial $h \in A$ is a zero divisor in $A/I$ or not.
Is there a computer algebra ...
- 21
5
votes
2
answers
640
views
Matroids relaxations of a given matroid
Let $\mathcal{M}$ be a rank-$d$ matroid on $[n]$. Say a matroid $\mathcal{N}$ is a relaxation of $\mathcal{M}$ if $\mathrm{rank}(\mathcal{N})=d$, $\mathrm{groundset}(\mathcal{N})=[n]$, and every ...
- 902