Elliptic Curve Rank Leaderboard

Rank-15 records in naive height, log|D| and Faltings: h=118.7702 (beats 120.6418), log|D|=108.6915 (beats 110.5171), Faltings=7.8218 (beats 7.9761); log N=100.5253 is not a record. Rank is exactly 15, not only a lower bound: ellrank returns [15,15,0]. Mestre rank-12 quartic family on the integer sextuple {-114,-112,44,55,57,70} at T=176: q(x,T)=p6(x-T)p6(x+T)=g^2-r, g monic of degree 6, deg r=4, curve = Jacobian of y^2=r(x). Found by enumerating the Mestre locus 12*p5=5*p2*p3 exhaustively over the integers rather than parametrizing it: centering (p1=0) and setting S=a1+a2+a3+a4, A_k=sum_{i<=4} a_i^k, f=a5*a6, the condition becomes a quadratic in f, -30*S*f^2 + (35*S^3-15*A2*S+10*A3)*f + (12*A5-5*A2*A3+5*A2*S^3-5*A3*S^2-7*S^5) = 0, with a5,a6 the roots of z^2+S*z+f=0 -- two integer square roots per 4-subset, so all 59479 primitive centered sextuples with |a_i|<=500 come out of 4.21e10 subsets in 87 s on 24 threads. Each was scanned over T=m/n (n<=8, m/n capped per sextuple by exact height), gating on exact naive height or log|D| already under the standing record before any Mestre-Nagao scoring: record fibres are downward height fluctuations at fixed T-scale, not top Nagao scorers. Points from hyperellratpoints on the quartic plus an ellratpoints integral sweep, carried to the Jacobian by x -> x0+1/w at a base point x0 (making the leading coefficient r(x0)=y0^2 square); 15 independent integral points, screened by Neron-Tate pairing then settled exactly by ellrank, which supplied the 15th point here (the cheap search scored this fibre 14) and capped nearly all other candidates rigorously at exactly 13 or 14. The sextuple sits in the two-zero-sum-cubics-with-equal-product slice ({-114,44,70} and {-112,55,57}, both of product -351120), the same slice as #235 -- the gain is exhaustive coverage at this scale, not a new family. The sibling equal-sum-of-squares branch looks degenerate: r comes out even with zero discriminant in every case sampled. Found with Claude Opus 5 (Anthropic).