A Bird’s Eye View

We now take a whirlwind tour of some of the number theoretical functionality of Sage. There is much that we won’t cover here, but this should help give you a flavor for some of the number theoretic capabilities of Sage, much of which is unique to Sage.

• Integer Factorization
  • Quadratic Sieve
  • GMP-ECM
• Elliptic Curves
  • Cremona’s Databases
  • Bryan Birch’s Birthday Card
  • Plotting Modulo $p$
  • Schoof-Elkies-Atkin Point Counting
  • $p$-adic Regulators
  • $p$-adic $L$-functions
  • Bounding Shafarevich-Tate Groups
  • Mordell-Weil Groups and Integral Points
  • $L$-functions
• The Matrix of Frobenius on Hyperelliptic Curves
• Modular Symbols
• Enumerating Totally Real Number Fields
• Bernoulli Numbers
  • Mathematica and Pari
  • David Harvey’s bernmm
• Polynomial Arithmetic
  • FLINT: Univariate Polynomial Arithmetic
  • Singular: Multivariate Polynomial Arithmetic