Infinite families of elliptic curves with high rank and prescribed torsion

Maintained by Andrej Dujella, University of Zagreb

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

G(T) = sup {rank E(Q(t)) : torsion group of elliptic curve E over Q(t) is T},

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

In the following two tables the best known lower bounds for G(T) and C(T) are given. If C(T) > G(T), it means that the current record for C(T) comes from a parametrization by rational points of some elliptic curves with positive rank.

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    T         G(T)>=             Author(s)
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    0           18        Elkies (2006) 
   Z/2Z         11        Elkies (2009)
   Z/3Z          7        Elkies (2007)
   Z/4Z          5        Kihara (2004), Elkies (2007), Dujella - Peral - Tadic (2014)		 
   Z/5Z          3        Lecacheux (2001), Eroshkin (2009), MacLeod (2014)
   Z/6Z          3        Lecacheux (2001), Kihara (2006), Eroshkin (2008), Woo (2008), 
                          Dujella - Peral (2012), MacLeod (2014)	 	 
   Z/7Z          1        Kulesz (1998), Lecacheux (2003), Rabarison (2008), 
                          Harrache (2009), MacLeod (2014)
   Z/8Z          2        Dujella - Peral (2012), MacLeod (2013) 
   Z/9Z          0        Kubert (1976)
   Z/10Z         0        Kubert (1976)  
   Z/12Z         0        Kubert (1976)
Z/2Z × Z/2Z      7        Elkies (2007)
Z/2Z × Z/4Z      4        Dujella - Peral (2012)
Z/2Z × Z/6Z      2        Dujella - Peral (2012), MacLeod (2013)
Z/2Z × Z/8Z      0        Kubert (1976) 
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___________________________________________________________________________________________________

    T         C(T)>=             Author(s)
___________________________________________________________________________________________________

    0           19        Elkies (2006) 
   Z/2Z         11        Elkies (2007)
   Z/3Z          7        Elkies (2007)
   Z/4Z          6        Elkies (2007) 		 
   Z/5Z          4        Eroshkin (2009)
   Z/6Z          5        Eroshkin (2009)	 	 
   Z/7Z          2        Lecacheux (2003), Elkies (2006), Rabarison (2008), Harrache (2009)
   Z/8Z          3        Dujella - Peral (2012) 
   Z/9Z          1        Atkin - Morain (1993), Kulesz (1998), Rabarison (2008), 
                          Gasull - Manosa - Xarles (2010)
   Z/10Z         1        Atkin - Morain (1993), Kulesz (1998), Rabarison (2008)
   Z/12Z         1        Suyama (1985), Kulesz (1998), Rabarison (2008)
Z/2Z × Z/2Z      8        Elkies (2007)
Z/2Z × Z/4Z      5        Eroshkin (2009)
Z/2Z × Z/6Z      3        Dujella - Peral (2013)
Z/2Z × Z/8Z      1        Atkin - Morain (1993), Kulesz (1998), Lecacheux (2002), 
                          Campbell - Goins (2003), Rabarison (2008) 	 	
___________________________________________________________________________________________________

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Old version of this tables (2006)


High rank elliptic curves with prescribed torsion

History of elliptic curves rank records

High rank elliptic curves with prescribed torsion over quadratic fields