High rank elliptic curves with prescribed torsion

Maintained by Andrej Dujella, University of Zagreb

Let T be an admissible torsion group for an elliptic curve over the rationals. Define

B(T) = sup {rank(E(Q)) : torsion group of elliptic curve E over Q is T}.

The conjecture is that B(T) is unbounded for all T. In the following table we give the best known lower bounds for B(T).

______________________________________________________________________________________________________

    T         B(T)>=	              Author(s)
______________________________________________________________________________________________________

    0           28        Elkies (2006)

   Z/2Z         19        Elkies (2009)	

   Z/3Z         13        Eroshkin (2007,2008,2009)

   Z/4Z         12        Elkies (2006)  

   Z/5Z          8        Dujella - Lecacheux (2009), Eroshkin (2009) 

   Z/6Z          8        Eroshkin (2008), Dujella - Eroshkin (2008), Elkies (2008), 
                          Dujella (2008), Dujella - Peral (2012) 	

   Z/7Z          5        Dujella - Kulesz (2001), Elkies (2006), Eroshkin (2009,2011), 
                          Dujella - Lecacheux (2009), Dujella - Eroshkin (2009)

   Z/8Z          6        Elkies (2006), Dujella - MacLeod - Peral (2013) 

   Z/9Z          4        Fisher (2009) 

   Z/10Z         4        Dujella (2005,2008), Elkies (2006)    

   Z/12Z         4        Fisher (2008) 

Z/2Z × Z/2Z     15        Elkies (2009)

Z/2Z × Z/4Z      9        Dujella - Peral (2012)

Z/2Z × Z/6Z      6        Elkies (2006)

Z/2Z × Z/8Z      3        Connell (2000), Dujella (2000,2001,2006,2008), Campbell - Goins (2003), 
                          Rathbun (2003,2006,2013), Dujella - Rathbun (2006), 
                          Flores - Jones - Rollick - Weigandt - Rathbun (2007), Fisher (2009)
                                                   
______________________________________________________________________________________________________

Click on rank r to see the corresponding "record" curve(s) with torsion points and independent points P1, P2, ... , Pr of infinite order.


References:

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Infinite families of elliptic curves with high rank and prescribed torsion

History of elliptic curves rank records

High rank elliptic curves with prescribed torsion over quadratic fields