Low-level quadratic norm data for the X₁(18) descent #
The anti-diagonal quotient of the surviving cubic correspondence has the genus-two model
z² = 33w⁶ + 18w⁵ + 15w⁴ - 20w³ + 15w² + 2w + 1.
Its sextic admits a useful factorization over ℚ(√-2). In homogeneous
coordinates, put
A = r³ + 9r²s - rs² - s³,
B = -4r(r²-s²).
We prove the exact identity A² + 2B² = F, where F is the homogeneous
sextic. More strongly, for coprime r,s the common divisor of A and B
divides eight. Thus after removing the forced parity factor the two
conjugate factors are coprime away from the ramified prime above two. These
are unconditional inputs to the fixed imaginary-quadratic descent, not a
rational-point classification. This lower module also contains the direct
rational-point transport from the original order-eighteen sextic. It stays
independent of the later Eisenstein-cube correspondence so that the immutable
X₁(18) challenge import can use these checked inputs without an import cycle.
The anti-diagonal quotient map #
The anti-diagonal quotient coordinate.
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The quadratic coefficient of the quotient equation.
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The square-root coordinate obtained from the discriminant of the quadratic quotient equation.
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- One or more equations did not get rendered due to their size.
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The exceptional polynomial forced by a zero quotient denominator.
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The homogeneous exceptional polynomial.
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The exceptional quotient-denominator polynomial has no rational root; the certificate is complete projective enumeration modulo five.
The exceptional denominator cannot vanish on a noncuspidal rational point of the cubic correspondence.
The discriminant of the quotient quadratic is nine times the anti-diagonal sextic.
Exact rational-function identity producing the anti-diagonal quotient of the surviving cubic correspondence.
A nondegenerate point of the cubic correspondence maps to the anti-diagonal genus-two curve.
The polynomial covariance identity underlying the Möbius comparison with the order-eighteen sextic already used by the Tate-normal-form consumer.
Rational-point transport from the order-eighteen model #
The Möbius coordinate carrying the order-eighteen sextic to the anti-diagonal model.
Equations
- MazurTorsion.XOneEighteenDescent.orderEighteenToAntiDiagonalW x = (x - 1) / (x + 1)
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The corresponding ordinate scaling on the order-eighteen sextic.
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The exceptional abscissa x = -1 cannot occur on the rational
order-eighteen sextic.
The forward Möbius coordinate has the displayed rational inverse away from the exceptional abscissa.
The inverse-coordinate spelling of the sextic covariance identity.
Every rational point on the order-eighteen sextic maps, with the checked ordinate scaling, to a rational point on the anti-diagonal sextic.
A rational point on the original order-eighteen model supplies a nonexceptional anti-diagonal point together with the checked inverse abscissa. This is the point-level consumer of both Möbius identities.
The exact norm identity over ℤ[√-2].
Homogenizing at the canonical denominator recovers the integral sextic form.
For primitive homogeneous coordinates, the two coefficients of the quadratic norm have common divisor dividing eight.
The common divisor is exactly one for opposite parities and exactly eight for two odd primitive coordinates.
A primitive integral point on the anti-diagonal sextic supplies a quadratic norm equation whose two coefficients have gcd one or eight. This is the checked consumer of the factorization and support bound.
Every rational point on the anti-diagonal sextic has canonical primitive homogeneous coordinates satisfying the supported quadratic norm equation. This is the rational-point consumer used by the subsequent fixed-field square-class allocation.