j-invariant

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, <a href="/facts/Felix_Klein/WmL9guqK">Felix Klein</a>'s j-invariant or j function, regarded as a function of a <a href="/facts/Complex_analysis/hgCoQlH0">complex variable</a> τ, is a <a href="/facts/Modular_function/PcnbYR1f">modular function</a> of weight zero for <a href="/facts/Special_linear_group/rsahah9w">special linear group</a> SL(2, Z) defined on the <a href="/facts/Upper_half-plane/7fPqrowr">upper half-plane</a> of <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a>. It is the unique such function that is <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic</a> away from a simple pole at the <a href="/facts/Cusp_(singularity)/QH3fEGry">cusp</a> such that
</p>

j
        
          (
          
            e
            
              2
              π
              i
              
                /
              
              3
            
          
          )
        
        =
        0
        ,
        
        j
        (
        i
        )
        =
        1728
        =
        
          12
          
            3
          
        
        .
      
    
    {\displaystyle j\left(e^{2\pi i/3}\right)=0,\quad j(i)=1728=12^{3}.}

<p><a href="/facts/Rational_function/2nJGu0Ko">Rational functions</a> of j are modular, and in fact give all modular functions of weight 0. Classically, the j-invariant was studied as a parameterization of <a href="/facts/Elliptic_curve/60jvaHSA">elliptic curves</a> over 
  
    
      
        
          C
        
      
    
    {\displaystyle \mathbb {C} }
  
, but it also has surprising connections to the symmetries of the <a href="/facts/Monster_group/MlwslQ7H">Monster group</a> (this connection is referred to as <a href="/facts/Monstrous_moonshine/WbyFQ68w">monstrous moonshine</a>).
</p>

j-invariant open-in-new

j-invariant