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

9,2

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.

Links

Table of n, a(n) for n=9..52.

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

Example

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.

Prog

(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

Crossrefs

Sequence in context: A086904 A358906 A360869 * A291276 A358274 A217305

Adjacent sequences: A379917 A379918 A379919 * A379921 A379922 A379923

Keyword

nonn

Author

Xavier Roulleau, Jan 17 2025

Status

approved