curve #243

y² + xy + y = x³ − 791198747812844165197303241658x + 291678735985274857428612896086571996361540568

a-invariants: [1, 0, 1, -791198747812844165197303241658, 291678735985274857428612896086571996361540568]
rank (lower bound): ≥ 20
conductor (N): 123989445321322909065995690886185865732040186216447789402341303782551770
naive height: 218.2916
Faltings height: 16.1469
discriminant (Δ): -5054684863451019643038749147096969353867668542170076589556829874258223258043978504934937500
primes of bad reduction: 2, 3, 5, 17, 19, 23, 29, 31, 59, 1657, 204191283231710516050941879612935200616637742318196045393
regulator: 26922583187358235172.6849204969274253584348869249045696391195036218961397153261
submitted by: David Renshaw
last updated: 2026-07-02 02:02:39

Witness: 20 independent points

(436876547159399, 5422665630197378101065)
(375990611505449, -6881048738091909428565)
(475838366133299, -4789192526536875389535)
(309626103514289, 8739924684576584646255)
(579770808085271/4, -107344757768733870289755/8)
(-151670202942496, 20203737650008769711175)
(474836608215689, 4801033336095140780355)
(1399900845934769, 43903256069991280960095)
(413316049389764, 5938892007725126484555)
(-491678658233761, 23702995629264462353655)
(207182831653769, -11689705199855651038365)
(2826878413762139, -143684810483260713380445)
(972792119473619, 21037676494215690500235)
(733163671616639, -10280918119880512036695)
(1159271608488944, -30535585821516332183880)
(584393010154124, 5374644645235395266835)
(1001929050655889, -22466688030826232997945)
(382267730323742, 6714852110077627182165)
(467508117994889, 4895637428325170187555)
(3327907052188819, -185782396222315328318005)

Commentary

New curve of rank >= 20 and naive height 218.2916, beating the previous rank-20 record 223.3165 (#92). Found as the specialization T = 3895/6 of the Mestre/Fermigier rank-12 quartic family on the rational Mestre sextuple mestre_ais(u = -7/2, v = 3/2) (roots {-1455/4, 2955/4, 1437/2, -1149/4, -1851/4, -687/2}): q(x,T) = p6(x-T) p6(x+T) = g^2 - r with deg r = 4, curve = Jacobian of y^2 = r(x). Candidate surfaced by a staged Mestre-Nagao sieve (M=1500 stage 1 over T = m/n, m <= 5000, n <= 20; M=6000 rescore ranked it #1 of 3135 candidates under the rank-20 height bar, score 103.65 vs 98.36 for Fermigier's rank-22 curve in the same normalization). The 20 independent points were assembled from hyperellratpoints on the integral quartic model to height 1e7 plus the 12 injected family base points x = ai +/- T and an integer sweep ellratpoints(E, [1e12, 1]); independence certified by positive-definite Neron-Tate Gram matrix (eigenvalue margin 1e-7 at 90 digits).

last edited by David Renshaw at 2026-07-02 02:02:39 · history