Oscar.jl

julia> using Pkg
julia> Pkg.add("Oscar")
julia> using Oscar

julia> using Oscar
  ___   ____   ____    _    ____
 / _ \ / ___| / ___|  / \  |  _ \   |  Combining ANTIC, GAP, Polymake, Singular
| | | |\___ \| |     / _ \ | |_) |  |  Type "?Oscar" for more information
| |_| | ___) | |___ / ___ \|  _ <   |  Manual: https://docs.oscar-system.org
 \___/ |____/ \____/_/   \_\_| \_\  |  Version 1.2.0-DEV
julia> k, a = quadratic_field(-5)
(Imaginary quadratic field defined by x^2 + 5, sqrt(-5))

julia> zk = maximal_order(k)
Maximal order of Imaginary quadratic field defined by x^2 + 5
with basis AbsSimpleNumFieldElem[1, sqrt(-5)]

julia> factorizations(zk(6))
2-element Vector{Fac{AbsSimpleNumFieldOrderElem}}:
 -1 * -3 * 2
 -1 * (-sqrt(-5) - 1) * (-sqrt(-5) + 1)

julia> Qx, x = polynomial_ring(QQ, [:x1,:x2])
(Multivariate polynomial ring in 2 variables over QQ, QQMPolyRingElem[x1, x2])

julia> R = grade(Qx, [1,2])[1]
Multivariate polynomial ring in 2 variables over QQ graded by
  x1 -> [1]
  x2 -> [2]

julia> f = R(x[1]^2+x[2])
x1^2 + x2

julia> degree(f)
[2]

julia> F = free_module(R, 1)
Free module of rank 1 over R

julia> s = sub(F, [f*F[1]])[1]
Submodule with 1 generator
1 -> (x1^2 + x2)*e[1]
represented as subquotient with no relations.

julia> H, mH = hom(s, quo(F, s)[1])
(hom of (s, Subquotient of
1 -> e[1]
by
1 -> (x1^2 + x2)*e[1]), Map: H -> set of all homomorphisms from s to subquotient of Submodule with 1 generator
1 -> e[1]
by Submodule with 1 generator
1 -> (x1^2 + x2)*e[1])

julia> mH(H[1])
Map with following data
Domain:
=======
Submodule with 1 generator
1 -> (x1^2 + x2)*e[1]
represented as subquotient with no relations.
Codomain:
=========
Subquotient of Submodule with 1 generator
1 -> e[1]
by Submodule with 1 generator
1 -> (x1^2 + x2)*e[1]

julia> C = Polymake.polytope.cube(3);

julia> C.F_VECTOR
pm::Vector<pm::Integer>
8 12 6

julia> RP2 = Polymake.topaz.real_projective_plane();

julia> RP2.HOMOLOGY
pm::Array<topaz::HomologyGroup<pm::Integer> >
({} 0)
({(2 1)} 0)
({} 0)

[OSCAR]
    OSCAR -- Open Source Computer Algebra Research system, Version 1.2.0-DEV,
    The OSCAR Team, 2024. (https://www.oscar-system.org)
[OSCAR-book]
    Wolfram Decker, Christian Eder, Claus Fieker, Max Horn, Michael Joswig, eds.
    The Computer Algebra System OSCAR: Algorithms and Examples,
    Algorithms and Computation in Mathematics, Springer, 2024.

@misc{OSCAR,
  key          = {OSCAR},
  organization = {The OSCAR Team},
  title        = {OSCAR -- Open Source Computer Algebra Research system,
                  Version 1.2.0-DEV},
  year         = {2024},
  url          = {https://www.oscar-system.org},
  }

@book{OSCAR-book,
  editor = {Decker, Wolfram and Eder, Christian and Fieker, Claus and Horn, Max and Joswig, Michael},
  title = {The {C}omputer {A}lgebra {S}ystem {OSCAR}: {A}lgorithms and {E}xamples},
  year = {2024},
  publisher = {Springer},
  series = {Algorithms and {C}omputation in {M}athematics},
  volume = {32},
  edition = {1},
  url = {https://link.springer.com/book/9783031621260},
  month = {8},
  issn = {1431-1550},
}