A332688 Reduced division polynomials associated with elliptic curve y^2 = x^3 + 17 and multiples of point (-2, 3).

0, 1, 1, -5, -23, 181, 1740, -8749, -1247497, -33863035, 4642247999, 675108007991, -68049162697560, -39309850524879431, 4258387436512005121, 10156867585318107377275, 4534892371184114325167497, -27717062339512733060005291331, -41962430757987463447351232692380

Offset

0,4

Comments

For the elliptic curve y^2 = x^3 + 17, the multiples of the point (-2, 3) are (u(n)/a(n)^2, v(n)/a(n)^3) as rationals.

Links

Table of n, a(n) for n=0..18.

LMFDB, Elliptic Curve 10404.f2 (Cremona label 10404c1).

Formula

a(n) = -a(-n), a(n+4)*a(n-3) = -23*a(n+3)*a(n-2) + 348*a(n+1)*a(n) for all n in Z.

a(n+12)*a(n-12) = (a(12)/a(6))^2 * a(n+6)*a(n-6) - a(18)/a(6) * a(n)^2 for all n in Z.

Example

a(4) = -23 and the point multiple 4*(-2,3) = (752/(-23)^2, 54239/(-23)^3).

Prog

(PARI) {a(n) = sign(n) * subst(elldivpol(ellinit([0, 17]), abs(n)), x, -2) / (if(n%2, 1, 6) * 2^(n^2\3) * 3^(n^2\4))};
(PARI) {a(n) = my(v, s); s=sign(n); n=abs(n); if(n==0, return(0)); v = concat([1, 1, -5, -23, 181, 1740, -8749], vector(max(0, n-7))); for(k=8, n, v[k] = (-23*v[k-1]*v[k-6] +348*v[k-3]*v[k-4])/v[k-7]); s*v[n]};
(PARI) {a(n) = my(v, s); s=sign(n); n=abs(n); if(n==0, return(0)); v = concat([1, 1, -5, -23], vector(max(0, n-4))); for(k=5, n, v[k] = (if(k%2, 3, 1)*v[k-1]*v[k-3] +if(k%3==2, 10, 5)*v[k-2]^2)/v[k-4]); s*v[n]};
(PARI) {a(n) = my(v, s); s=sign(n); n=abs(n); if(n==0, return(0)); v = concat([1, 1, -5, -23, 181], vector(max(0, n-5))); for(k=6, n, v[k] = (-if(k%3==1, 10, 5)*v[k-1]*v[k-4] +23*v[k-2]*v[k-3])/v[k-5]); s*v[n]};

Crossrefs

Sequence in context: A116151 A371651 A305127 * A198026 A198093 A197792

Adjacent sequences: A332685 A332686 A332687 * A332689 A332690 A332691

Keyword

sign

Author

Michael Somos, Feb 19 2020

Status

approved