A379920 Number of irreducible conic curves containing 6 points of a cyclic order n-torsion subgroup of an elliptic curve.

1, 2, 7, 13, 36, 67, 113, 196, 312, 455, 693, 984, 1353, 1869, 2508, 3261, 4284, 5478, 6898, 8684, 10780, 13174, 16146, 19516, 23381, 27976, 33201, 39041, 45936, 53601, 62187, 72048, 83028, 95109, 108927, 124068, 140749, 159467, 179998, 202321, 227304, 254380, 283844, 316360

Offset: 9

Author

Xavier Roulleau, Jan 17 2025

Keywords

• nonn

Comments

For n < 9, there are no such curves.

There are precisely 7 primes in this sequence, namely

a(10)=2, a(11)=7, a(12)=13, a(14)=67, a(15)=113, a(36)=39041, a(63)=907237.

Examples

			For n=9, there is a unique irreducible conic that contains 6 points in a cyclic order n torsion subgroup of an elliptic curve, and for n=11 there are 7 such conics.
		

Links

• David Broadhurst and Xavier Roulleau, Number of partitions of modular integers, arXiv:2502.19523 [math.NT], 2025. See pp. 3, 19.

Programs

Magma

sq:=[];
for NN in [9..30] do
G:=Integers(NN);
SG:={q: q in G};
QNT:=Subsets(SG,5);
QNT:={q join {-(&+ q)} : q in QNT | not -(&+ q) in q};
TRS:=Subsets(SG,3);
TRS:={q : q in TRS|&+q eq 0};
QNT:={q :q in QNT| not #{u : u in TRS| u subset q} ge 1};
Append(~sq,#QNT);
end for;
sq;

PARI

{a(n)=[(n-6)*(n^4-19*n^3+121*n^2-384*n+840),(n-1)*(n-4)*(n-5)*(n-7)*(n-8),(n-2)*(n-4)*(n-8)*(n^2-11*n+25),(n-3)*(n^4-22*n^3+169*n^2-588*n+1200)][gcd(n,6)%6+1]/6!;} \\ David Broadhurst, Jan 17 2025

Formula

G.f.: x^9*(1 + x + 3*x^2 + 2*x^3 + 12*x^4 + 14*x^5 - 3*x^6 - x^7 + 7*x^8)/((1 - x)*(1 - x^2)^2*(1 - x^3)^2*(1 - x^6)) \\ David Broadhurst, Jan 17 2025