#!/usr/bin/env sage -python
from sage.all import EllipticCurve

from demo_package import greet


def describe_curve() -> None:
    """Construct an elliptic curve and print basic arithmetic information."""
    E = EllipticCurve([0, 0, 0, -1, 1])  # y^2 = x^3 - x + 1
    print(f"Elliptic curve E: {E}")
    print(f"Discriminant: {E.discriminant()}")
    print(f"j-invariant: {E.j_invariant()}")

    rank = E.rank()
    print(f"Rank: {rank}")

    generators = E.gens()
    if generators:
        P = generators[0]
        print(f"Generator: {P}")
        for n in range(1, 6):
            multiple = n * P
            print(f"{n} * P = {multiple}")
    else:
        print("Curve has no generators over Q.")


def main() -> None:
    print(greet("Sage enthusiast"))
    describe_curve()


if __name__ == "__main__":
    main()