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This is supposed to be a highlevel overview and list of functionality that is easily available from the standard SAGE interface. In particular, don't list functionality that is only available, e.g., by directly calling GAP or other systems. Also, this is aimed at people who have never heard of programs like GAP, Singular, etc., and don't care how SAGE does things  they just want to know exactly *what* SAGE can do. 

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* Computing a Groebner basis is fast because of the SINGULAR computer algebra system.  * Fast computation of Groebner basis. 
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== Crypto ==  == Cryptography == 
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* The notebook is a useful tool for basic math education because of its flexible visualization/output capabilities.  * The SAGE notebook (a graphical interface) is a useful tool for basic math education because of its flexible visualization/output capabilities. 
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* SAGE provides interpreter interfaces to Axiom, CoCoA, GAP, KASH, Macaulay2, Magma, Maple, Mathematica, Matlab, Maxima, MuPAD, Octave, and Singular. * SAGE provides C/C++library interfaces to NTL, PARI, Linbox, and mwrank. 
* Interpreter interfaces to Axiom, CoCoA, GAP, KASH, Macaulay2, Magma, Maple, Mathematica, Matlab, Maxima, MuPAD, Octave, and Singular. * C/C++library interfaces to NTL, PARI, Linbox, and mwrank. 
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* The reduced row echelon form of e.g. dense 20,000x20,000 matrices over GF(2) can be computed in seconds and 50MB of RAM. * Computation of reduced row echelon forms of sparse matrices is supported. 
* Compute the reduced row echelon form of e.g. dense 20,000x20,000 matrices over GF(2) in seconds and 50MB of RAM. * Computation of reduced row echelon forms of sparse matrices. * Fast matrix multiplication, characteristic polynomial and echelon forms of dense matrices over QQ. 
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* Compute MordellWeil groups of (many) elliptic curves using both invariants and algebraic 2descents. * A wide range of number theoretic functions, e.g., euler_phi, primes enumeration, sigma, tau_qexp, etc. * Compute the number of points on an elliptic curve modulo p for all primes p less than a million in seconds. * Optimized implementation of the SEA point counting algorithm for counting points modulo p when p is large. * An optimized modern quadratic Sieve for factoring integers n = p*q. * Modular symbols for general weight, character, Gamma1, and GammaH. * Modular forms for general weight >= 2, character, Gamma1, and GammaH. 

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* Fast arithmetic and special functions * Matrix and vector arithmetic, QR decomposition, system solving 

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* SAGE provides 2d plotting functionality similar to Mathematica's. * SAGE provides limited 3d plotting via an included ray tracer. 
What SAGE Can Do
This is supposed to be a highlevel overview and list of functionality that is easily available from the standard SAGE interface. In particular, don't list functionality that is only available, e.g., by directly calling GAP or other systems. Also, this is aimed at people who have never heard of programs like GAP, Singular, etc., and don't care how SAGE does things  they just want to know exactly *what* SAGE can do.
Calculus
Commutative Algebra
 Fast computation of Groebner basis.
Cryptography
 Classical ciphers are well supported.
Elementary Education
 The SAGE notebook (a graphical interface) is a useful tool for basic math education because of its flexible visualization/output capabilities.
Finite Fields
 Basic arithmetic over finite extension fields is fast because of the Givaro library.
Graphical Interface
Group Theory
Interfaces
 Interpreter interfaces to Axiom, CoCoA, GAP, KASH, Macaulay2, Magma, Maple, Mathematica, Matlab, Maxima, MuPAD, Octave, and Singular.
 C/C++library interfaces to NTL, PARI, Linbox, and mwrank.
Linear Algebra
 Compute the reduced row echelon form of e.g. dense 20,000x20,000 matrices over GF(2) in seconds and 50MB of RAM.
 Computation of reduced row echelon forms of sparse matrices.
 Fast matrix multiplication, characteristic polynomial and echelon forms of dense matrices over QQ.
Number Theory
 Compute MordellWeil groups of (many) elliptic curves using both invariants and algebraic 2descents.
 A wide range of number theoretic functions, e.g., euler_phi, primes enumeration, sigma, tau_qexp, etc.
 Compute the number of points on an elliptic curve modulo p for all primes p less than a million in seconds.
 Optimized implementation of the SEA point counting algorithm for counting points modulo p when p is large.
 An optimized modern quadratic Sieve for factoring integers n = p*q.
 Modular symbols for general weight, character, Gamma1, and GammaH.
Modular forms for general weight >= 2, character, Gamma1, and GammaH.
Numerical Computation
 Fast arithmetic and special functions
 Matrix and vector arithmetic, QR decomposition, system solving
padic Numbers
Plotting
 SAGE provides 2d plotting functionality similar to Mathematica's.
 SAGE provides limited 3d plotting via an included ray tracer.