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}.

In the following table we give the best known lower bounds for B(T). 

______________________________________________________________________________________________________

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

    0           29        Elkies - Klagsbrun (2024)	

   Z/2Z         20        Elkies - Klagsbrun (2020)	

   Z/3Z         15        Elkies - Klagsbrun (2020)

   Z/4Z         13        Elkies - Klagsbrun (2020) 

   Z/5Z          9        Klagsbrun (2020) 

   Z/6Z          9        Klagsbrun (2020), Voznyy (2020)	

   Z/7Z          6        Klagsbrun (2020)

   Z/8Z          6        Elkies (2006), Dujella - MacLeod - Peral (2013), Voznyy (2021) 

   Z/9Z          4        Fisher (2009), van Beek (2015), Dujella - Petricevic (2021), 
                          Dujella - Petricevic - Rathbun (2022) 

   Z/10Z         4        Dujella (2005,2008), Elkies (2006), Fisher (2016)    

   Z/12Z         4        Fisher (2008) 

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

Z/2Z × Z/4Z      9        Dujella - Peral (2012,2019), Klagsbrun (2020)

Z/2Z × Z/6Z      6        Elkies (2006), Dujella - Peral - Tadic (2015), Dujella - Peral (2020)

Z/2Z × Z/8Z      3        Connell (2000), Dujella (2000,2001,2006,2008), Campbell - Goins (2003), 
                          Rathbun (2003,2006,2013,2022), Dujella - Rathbun (2006), 
                          Flores - Jones - Rollick - Weigandt - Rathbun (2007), Fisher (2009), 
                          AttarBashi - Rathbun - Voznyy (2022), 
			  AttarBashi - Fisher - Rathbun - Voznyy (2022), 
			  AttarBashi - Fisher - Voznyy (2022)         
                                                   
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Click on rank r to see the corresponding "record" curve(s) with torsion points and independent points P1, P2, ... , Pr of infinite order.

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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

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