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Numbers k for which rank of the elliptic curve y^2 = x^3 - 432*k^2 is 5.
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%I #39 Jan 25 2023 09:08:36

%S 489489,525698,526535,763002,903210,1423214

%N Numbers k for which rank of the elliptic curve y^2 = x^3 - 432*k^2 is 5.

%F A060838(a(n)) = 5.

%o (PARI) is(n)=my(c=prod(i=1, #f~, f[i, 1]^(f[i, 2]\3)), r=n/c^3, E=ellinit([0, 16*r^2]), eri=ellrankinit(E), mwr=ellrank(eri), ar); if(r<489489, return(0)); if(mwr[1]>5 || mwr[2]<5, return(0)); ar=ellanalyticrank(E)[1]; if(ar<2, return(0)); for(effort=1, 99, mwr=ellrank(eri, effort); if(mwr[1]>5 || mwr[2]<5, return(0), mwr[1]==5 && mwr[2]==5, return(1))); Str("unknown; ",ar==5," under BSD conjecture") \\ _Charles R Greathouse IV_, Jan 25 2023

%Y Subsequence of A159843.

%Y Cf. A060748, A060838, A309960 (rank 0), A309961 (rank 1), A309962 (rank 2), A309963 (rank 3), A309964 (rank 4).

%K nonn,more

%O 1,1

%A Maksym Voznyy and _Charles R Greathouse IV_, Jan 25 2023