curve #234

y² + xy + y = x³ − 2770478907376168032636869x + 1775138479573443551078307813696866276

a-invariants: [1, 0, 1, -2770478907376168032636869, 1775138479573443551078307813696866276]
rank (lower bound): ≥ 19
conductor (N): 121367195540260815785780563003076660369683539545326996733038410 ★ record for rank ≥ 19
naive height: 180.4570
Faltings height: 12.7393
discriminant (Δ): -325021758664267435353269545802736616838357964735913965003055852319441700
primes of bad reduction: 2, 3, 5, 7, 17, 7662261289645586443997884141, 4436864266498037047431696210593
regulator: 715538233792403396.6191825552383393257158767106899353176103967820819382092534
submitted by: RoyManami
last updated: 2026-07-01 22:30:41

Witness: 19 independent points

(939237708948, -39562762581743552)
(1128794739009, -293465032196013458)
(921248768118, 68566127545385158)
(1513488732693, 1024167210723639748)
(1302979350498, -614330875561576877)
(983923487169, -41724359928135728)
(976260793788, 29802115059241003)
(444611621769, 794507147074906408)
(1194706365594, -412870946793845453)
(472461354918, -756080230805621117)
(957288103968, -15852600561045077)
(441115178406, 799294168677977731)
(2456535768288, -3129456458737080917)
(1134305072523, 303349975608016648)
(1175405086086, 377645080063855222)
(982431954861, -39342857401778228)
(1401568389501, -803340781131078404)
(3567736384359/4, 933938344059977861/8)
(7999856558043, -22171701369808730492)

Commentary

Rank-19 height AND conductor record: h=180.457 (beats 190.312), log N=142.954 (beats 146.224). New 3-parameter sub-family of Mestre locus 12*p5=5*p2*p3: six roots = union of two depth-2 cubics x^3+px+q, x^3+rx+s with (p-r)(q-s)=0 (equal product), disjoint from Mestre (u,v) slice. Sextuple {0,203,204,254,370,523}, T=1303/3; base-point injection + Neron-Tate independence (19 indep).

last edited by RoyManami at 2026-06-26 01:21:19 · history