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Number of elements of the finite group defined by the elliptic curve y^2 = x^3 + x + 1 (mod prime(n)).
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%I #20 Aug 02 2023 10:20:34

%S 3,4,9,5,14,18,18,21,28,36,33,48,35,34,60,58,63,50,56,59,72,86,90,100,

%T 97,105,87,105,123,125,126,128,126,126,136,154,171,189,144,172,180,

%U 190,217,201,222,218,223,244,228,232,237,262,220,282,249,260,294,274,256,289

%N Number of elements of the finite group defined by the elliptic curve y^2 = x^3 + x + 1 (mod prime(n)).

%C Any elliptic curve can be written as a plane algebraic curve defined by an equation of the form: y^2 = x^3 + ax + b.

%H Robin Visser, <a href="/A192334/b192334.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EllipticCurveGroupLaw.html">Elliptic Curve Group Law</a>.

%F a(n) = 1+A098514(n), which does not count (oo,oo). - _R. J. Mathar_, Jun 29 2011

%e Example : a(8) = 21 because, for n = 8, prime(8) = 19, and the field F(19) has 21 elements :(19, 1), (10, 2), (14, 2), (7, 3), (15, 3), (16, 3), (5, 6), (9, 6), (2, 7), (13, 8), (13, 11), (2, 12), (5, 13), (9, 13), (7, 16), (15, 16), (16, 16), (10, 17), (14, 17), (19,18) and the point at infinity.

%p with(numtheory):for n from 1 to 100 do:p:=ithprime(n):it:=0:for y from 1 to p do:for x from 1 to p do:z:=x^3+x+1:z1:=irem(z,p):z2:=irem(y^2,p):if z1=z2 then it:=it+1:else fi:od:od: printf(`%d, `,it+1): od:

%t Reap[ For[ n=1, n <= 60 , n++, p = Prime[n]; it=0; For[ y=1 , y <= p , y++, For[ x=1 , x <= p , x++, z = x^3+x+1; z1 = Mod[z, p]; z2 = Mod[y^2, p]; If[ z1 == z2 , it = it+1]]]; Sow[it+1]]][[2, 1]](* _Jean-François Alcover_, Jun 11 2012, translated from Maple *)

%K nonn

%O 1,1

%A _Michel Lagneau_, Jun 28 2011