Quotient and arithmetic descent identities for X₁(18) #
The order-three automorphism of the order-eighteen sextic has invariant functions
u = (x³ - 3x + 1)/(x(x-1)),
v = y/(x(x-1)),
which satisfy
v² = u² - 4u + 12.
This file parameterizes that rational conic, identifies the degree-three fiber as the cyclic cubic
z³ - uz² + (u-3)z + 1,
and records its orbit factorization and square discriminant. After the conic parameter is introduced, the cleared cubic has discriminant
(t²+3)²(7t²+6t+3)².
The two quadratic factors are explicit norms in Eisenstein coordinates.
Writing ω²+ω+1=0, the second is
N((3+ω)m + (1-ω)n) = 7m²+6mn+3n²,
and N(3+ω)=7. We prove these norm identities, multiplicativity, two
primitive-form Bézout identities, and the fact that the two norm forms
cannot both be divisible by 7 for coprime integer parameters.
These are algebraic and local prerequisites for the classical
π = 3+ω descent. They do not construct the induced endomorphism of the
Jacobian, prove π-surjectivity on its Mordell--Weil group, determine its
torsion, or classify rational points.
The rational conic quotient #
The slope used to recover the conic parameter away from u=1.
Equations
- MazurTorsion.XOneEighteenDescent.conicSlope u v = (v - 3) / (u - 1)
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The cyclic cubic fiber #
The exceptional quotient fiber u=1 has no rational abscissa.
The invariant abscissa of a noncuspidal point satisfies its cubic fiber equation.
A noncuspidal rational abscissa never lies over the exceptional
quotient value u=1.
Product over the three-element Möbius orbit, viewed as a degree-three norm polynomial.
Equations
- One or more equations did not get rendered due to their size.
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The orbit norm is exactly the cyclic cubic fiber polynomial.
The standard discriminant formula for a cubic
a z³ + b z² + c z + d.
Equations
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Clearing the conic denominator gives clearedFiber.
A root of the parameterized fiber gives a root of the cleared cubic.
The homogeneous cubic obtained by writing the conic parameter as
t=m/n.
Equations
Instances For
Substitution t=m/n turns the cleared cubic into its homogeneous
form after multiplication by n².
A root of the cleared rational-parameter cubic is a root of the primitive homogeneous cubic attached to its canonical fraction.
The homogeneous cubic discriminant is the product of the squares of the two quadratic norm forms.
Every rational noncuspidal point on the sextic admits the explicit conic parameter and is a root of the resulting denominator-free cubic.
Canonical numerator-denominator coordinates sharpen the preceding statement to primitive integer parameters.
The coefficient discriminant of the cleared cubic factors into two squared quadratic forms.
Eisenstein-coordinate norm identities #
The coordinate norm is multiplicative.
Multiplication by the conjugate gives the rational integer norm.
Conjugation preserves the Eisenstein norm.
The Eisenstein prime above 7, represented by π=3+ω.
Equations
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The norm of π=3+ω is 7.
The conjugate prime is conj(π)=2-ω.
The coordinate factorization of the rational prime 7 as
π·conj(π).
Divisibility by π in the coordinate ring.
Equations
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A factor of π forces a factor of 7 in the norm.
The two primitive quadratic norm forms.
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For integer homogeneous parameters, the two factors in the cubic discriminant are exactly the two named norm forms.
A Bézout-style identity eliminating the denominator parameter.
For primitive integer parameters, the two Eisenstein norm forms
cannot both be divisible by the rational prime below π.
The complete elementary descent package obtained here from a
noncuspidal rational point: primitive homogeneous parameters, the
homogeneous cubic equation, and the local exclusion of a simultaneous
factor of 7 in its two Eisenstein norm factors.