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Magma is a software package for solving computationally hard problems in algebra, number theory, geometry and combinatorics. It provides a mathematically rigorous environment for computing with algebraic, number-theoretic, combinatoric and geometric objects. Magma can be compared with Mathematica, Maple or Matlab. It strength is in having built-in cutting edge and the most sophisticated algorithms known. Magma runs at the Mathematics department on the machine modular.fas.harvard.edu, for which you need a separate account. More Magma (and Pari ) documentation can be found here .
Snippets of code examples: (watching simple examples is a good way to learn and appreciate the language). Some of the examples have been taken from the Magma documentation. More examples are available here .

 amicable.magma Amicable numbers, how to make loops complex.magma Complex numbers, calculation with arbitrary precision factorisation.magma Factor integers, with report on used time factorpolynomial.magma Factor polynomials over finite fields factorpolynomialz.magma Factor polynomials over integer ring simplelist.magma Building a simple list lists.magma Doing operations on Lists primes.magma Print list of primes up to 100000 farey.magma Factor polynomials over integer ring, how to define functions finitefield.magma defining polynomial for a finite field groebner.magma get the Groebner Basis of an ideal group.magma finitely presented group parametrized by parameters in relations, how to make lists infinitegroup.magma defining an infinite finitely presented group knapsack.magma Solving the knapsack problem, simple programming example matrixgroup.magma Working with Matrix groups permutation.magma Defining a permutation group generated by a few permutations subgroup.magma finding the subgroup lattice of a group primenumber.magma Illustration of the prime number theorem, example with formatted output sumsquare.magma Writing number as a sum of two squares

 Simplicity, Clarity, Generality B.W. Kernighan, R. Pike, in "The Practice of Programming".