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# 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

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