A197548 Rank of quartic elliptic curve y^2 = 5*x^4 + 4*n.

1, 1, 0, 2, 0, 0, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 0, 0, 1, 1, 1, 1, 1, 2, 0, 1, 1, 2, 1, 2, 1, 0, 0, 2, 1, 1, 0, 1, 2, 0, 1, 1, 1, 1, 1, 1, 2, 0, 0, 0, 0, 2, 0, 1, 1, 2, 2, 0, 0, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 0, 1, 2

Offset

1,4

Comments

If a(n)=0, it means that the number of rational points on curve y^2=5*x^4+4*n is finite; if a(n)>0, the number of rational points is infinite. The value of the rank tells how many points of infinite order is necessary to generate complete infinite set of rational points of given curve.

The quintic trinomial of the form x^5-n*x+m has only finitely many cases such that is factorizable on quadratic and cubic factor with different Elkies coefficient n^5/m^4 if and only a(n)=0; if a(n)>0, then there are infinitely many solutions.

Links

Table of n, a(n) for n=1..100.

Prog

(Magma) for n := 1 to 100 do print([n, Rank(EllipticCurve([5, 0, 0, 0, 4*n]))]); end for; (*Max Alekseyev*)

Crossrefs

Cf. A193524, A193528, A193531, A193533, A193584, A197547.

Sequence in context: A052154 A360002 A039977 * A029403 A007706 A241069

Adjacent sequences: A197545 A197546 A197547 * A197549 A197550 A197551

Keyword

nonn

Author

Artur Jasinski, Oct 16 2011

Status

approved