A197547 Rank of quartic elliptic curve y^2 = 5*x^4 - 4*n.

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

Offset

1,2

Comments

According MAGMA ranks for n={33,67,97} are only lower bounds (maybe accurate but can be higher).

If a(n)=0 that means that the number of rational points on curve y^2=4*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 finite many rational solutions 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 1000 do print([n, Rank(EllipticCurve([5, 0, 0, 0, -4*n]))]); end for; (*Max Alekseyev*)

Crossrefs

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

Sequence in context: A057594 A259029 A286128 * A239723 A229541 A015488

Adjacent sequences: A197544 A197545 A197546 * A197548 A197549 A197550

Keyword

nonn

Author

Artur Jasinski, Oct 16 2011

Status

approved