Normalize the cusp y^2=x^3 with exact conductor evidence¶
Continuously checked
This is the exact source run by make test-examples and the ordinary Python test gate.
Source: examples/36_affine_normalization.py
# SPDX-License-Identifier: MIT
# Copyright (c) 2026 Benjamin Antieau and Deven Manam
"""Normalize the cusp y^2=x^3 with exact conductor evidence."""
from noether import Q
from noether.commalg import PolynomialRing
def main() -> None:
ring = PolynomialRing(Q, ["x", "y"])
x, y = ring.generators()
cusp = ring.ideal([y**2 - x**3])
normalization = cusp.normalization()
normalization.verify()
target = normalization.normal_algebra.base_ring
t = target.generator(0)
assert normalization.coordinate_images == [t**2, t**3]
assert normalization.source_conductor == ring.ideal([x, y])
assert not normalization.is_normal
print(
f"cusp normalization: x -> t^2, y -> t^3; proof={normalization.proof_kind}, conductor=(x,y)"
)
if __name__ == "__main__":
main()