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[parent] j-invariant (Definition)

Let $ E$ be an elliptic curve over $ \mathbb{Q}$ with Weierstrass equation:

$\displaystyle y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$
with coefficients $ a_i\in\mathbb{Q}$. Let:
$\displaystyle b_2$ $\displaystyle =$ $\displaystyle a_1^2+4a_2,$  
$\displaystyle b_4$ $\displaystyle =$ $\displaystyle 2a_4+a_1a_3,$  
$\displaystyle b_6$ $\displaystyle =$ $\displaystyle a_3^2+4a_6,$  
$\displaystyle b_8$ $\displaystyle =$ $\displaystyle a_1^2a_6+4a_2a_6-a_1a_3a_4+a_3^2a_2-a_4^2,$  
$\displaystyle c_4$ $\displaystyle =$ $\displaystyle b_2^2-24b_4,$  
$\displaystyle c_6$ $\displaystyle =$ $\displaystyle -b_2^3+36b_2b_4-216b_6$  

Definition 1       
  1. The discriminant of $ E$ is defined to be
    $\displaystyle \Delta=-b_2^2b_8-8b_4^3-27b_6^2+9b_2b_4b_6$
  2. The j-invariant of $ E$ is
    $\displaystyle j=\frac{c_4^3}{\Delta}$
  3. The invariant differential is
    $\displaystyle \omega=\frac{dx}{2y+a_1x+a_3}=\frac{dy}{3x^2+2a_2x+a_4-a_1y}$

Example:

If $ E$ has a Weierstrass equation in the simplified form $ y^2=x^3+Ax+B$ then

$\displaystyle \Delta=-16(4A^3+27B^2),\quad j=-\frac{1728(4A)^3}{\Delta}$

Note: The discriminant $ \Delta$ coincides in this case with the usual notion of discriminant of the polynomial $ x^3+Ax+B$.



"j-invariant" is owned by alozano.
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See Also: elliptic curve, bad reduction, modular discriminant, discriminant, the arithmetic of elliptic curves

Other names:  discriminant, $j$-invariant, j invariant
Also defines:  j-invariant, discriminant of an elliptic curve, invariant differential
Keywords:  j-invariant, discriminant, differential, elliptic curve

This object's parent.

Attachments:
the $j$-invariant classifies elliptic curves up to isomorphism (Theorem) by alozano
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Cross-references: coefficients, Weierstrass equation, elliptic curve
There are 9 references to this entry.

This is version 6 of j-invariant, born on 2003-08-07, modified 2005-03-01.
Object id is 4565, canonical name is JInvariant.
Accessed 9952 times total.

Classification:
AMS MSC: 14H52 (Algebraic geometry :: Curves :: Elliptic curves)
Pending Errata and Addenda
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