J-invariant - Revision history

Rational Point: fix subtle errors that have been creeping in

2025-02-25T23:26:33Z

fix subtle errors that have been creeping in

← Older revision		Revision as of 23:26, 25 February 2025
Line 1:		Line 1:
	The ''' ''j''-invariant''' of an elliptic curve in [[Weierstraß normal form]]		The ''' ''j''-invariant''' of an elliptic curve in [[Weierstraß normal form]]

	:<math>y^2=x^2+ax+b</math>		:<math>y^2=x^3+ax+b</math>

	is defined to be		is defined to be
Line 15:		Line 15:
	:<math>j=\frac{64a^3}{D}</math>		:<math>j=\frac{64a^3}{D}</math>

	in terms of the discriminant ''D'' of the cubic polynomial <math>x^2+ax+b</math>.		in terms of the discriminant ''D'' of the cubic polynomial <math>x^3+ax+b</math>.

Rational Point: in terms of the discriminant ''D'' of the cubic polynomial

2025-02-25T23:25:06Z

in terms of the discriminant ''D'' of the cubic polynomial

← Older revision		Revision as of 23:25, 25 February 2025
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	:<math>j=\frac{-110592a^3}{\Delta}</math>		:<math>j=\frac{-110592a^3}{\Delta}</math>

	in terms of the elliptic [[discriminant]] Δ <ref>Wolfram MathWorld: j-Invariant. https://mathworld.wolfram.com/j-Invariant.html</ref>.		in terms of the elliptic [[discriminant]] Δ <ref>Wolfram MathWorld: j-Invariant. https://mathworld.wolfram.com/j-Invariant.html</ref>, or,

			:<math>j=\frac{64a^3}{D}</math>

			in terms of the discriminant ''D'' of the cubic polynomial <math>x^2+ax+b</math>.

Rational Point: bad numbers and no refs

2025-01-03T16:58:03Z

bad numbers and no refs

← Older revision		Revision as of 16:58, 3 January 2025
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	or		or

	:<math>j=\frac{-~~432a~~^3}{\Delta}</math>		:<math>j=\frac{-110592a^3}{\Delta}</math>

	in terms of the elliptic [[discriminant]] Δ <ref>Wolfram MathWorld: j-Invariant. https://mathworld.wolfram.com/j-Invariant.html</ref>.		in terms of the elliptic [[discriminant]] Δ <ref>Wolfram MathWorld: j-Invariant. https://mathworld.wolfram.com/j-Invariant.html</ref>.

Rational Point: off by factor of 4

2025-01-03T16:45:02Z

off by factor of 4

← Older revision		Revision as of 16:45, 3 January 2025
Line 5:		Line 5:
	is defined to be		is defined to be

	:<math>j=\frac{~~1728a~~^3}{4a^3+27b^2}</math>		:<math>j=\frac{6912a^3}{4a^3+27b^2}</math>

	or		or

	:<math>j=\frac{-~~108a~~^3}{\Delta}</math>		:<math>j=\frac{-432a^3}{\Delta}</math>

	in terms of the elliptic [[discriminant]] Δ <ref>Wolfram MathWorld: j-Invariant. https://mathworld.wolfram.com/j-Invariant.html</ref>.		in terms of the elliptic [[discriminant]] Δ <ref>Wolfram MathWorld: j-Invariant. https://mathworld.wolfram.com/j-Invariant.html</ref>.

Rational Point: def

2025-01-03T16:40:10Z

def

New page

The ''' ''j''-invariant''' of an elliptic curve in [[Weierstraß normal form]]

:<math>y^2=x^2+ax+b</math>

is defined to be

:<math>j=\frac{1728a^3}{4a^3+27b^2}</math>

or

:<math>j=\frac{-108a^3}{\Delta}</math>

in terms of the elliptic [[discriminant]] Δ <ref>Wolfram MathWorld: j-Invariant. https://mathworld.wolfram.com/j-Invariant.html</ref>.