cp's OEIS Frontend

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

%I A379920 #51 Mar 06 2025 08:48:44
%S A379920 1,2,7,13,36,67,113,196,312,455,693,984,1353,1869,2508,3261,4284,5478,
%T A379920 6898,8684,10780,13174,16146,19516,23381,27976,33201,39041,45936,
%U A379920 53601,62187,72048,83028,95109,108927,124068,140749,159467,179998,202321,227304,254380,283844,316360
%N A379920 Number of irreducible conic curves containing 6 points of a cyclic order n-torsion subgroup of an elliptic curve.
%C A379920 For n < 9, there are no such curves.
%C A379920 There are precisely 7 primes in this sequence, namely
%C A379920 a(10)=2, a(11)=7, a(12)=13, a(14)=67, a(15)=113, a(36)=39041, a(63)=907237.
%H A379920 David Broadhurst and Xavier Roulleau, <a href="https://arxiv.org/abs/2502.19523">Number of partitions of modular integers</a>, arXiv:2502.19523 [math.NT], 2025. See pp. 3, 19.
%F A379920 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
%e A379920 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.
%o A379920 (Magma)
%o A379920 sq:=[];
%o A379920 for NN in [9..30] do
%o A379920 G:=Integers(NN);
%o A379920 SG:={q: q in G};
%o A379920 QNT:=Subsets(SG,5);
%o A379920 QNT:={q join {-(&+ q)} : q in QNT | not -(&+ q) in q};
%o A379920 TRS:=Subsets(SG,3);
%o A379920 TRS:={q : q in TRS|&+q eq 0};
%o A379920 QNT:={q :q in QNT| not #{u : u in TRS| u subset q} ge 1};
%o A379920 Append(~sq,#QNT);
%o A379920 end for;
%o A379920 sq;
%o A379920 (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
%K A379920 nonn
%O A379920 9,2
%A A379920 _Xavier Roulleau_, Jan 17 2025