curve #161

y² + xy + y = x³ − x² − 186119222080171360016717948x + 966784961134576267766275148260409718847

a-invariants: [1, -1, 1, -186119222080171360016717948, 966784961134576267766275148260409718847]
rank (lower bound): ≥ 18
conductor (N): 15293463740752553768990420489911754342865071901048558390 ★ record for rank ≥ 18
naive height: 193.0789
Faltings height: 13.9583
discriminant (Δ): 8844410691059589836574217469287328213356255539746089974415484198645460595507200
primes of bad reduction: 2, 3, 5, 13, 19, 23, 31, 37, 157, 163, 259627, 39576820317654329, 99173840155499332801
regulator: 1720477128584811.6188214221553592298019361587306052672826796241020503339218
submitted by: cocoxhuang
last updated: 2026-07-01 22:32:42

Witness: 18 independent points

(-13832435464763, -29910199873434594523)
(-15238574724275, 16259276081044689533)
(7127974835053, 1513322317013240093)
(8923742719093, 4065823273255454213)
(-15682513924187, 5350268149265678453)
(2246837286493, -23663221783492721707)
(7005303109877, 2596829909785016837)
(158755543668949/25, 800012588152647945433/125)
(-10537945303307, -41927054304229542907)
(267554946135157/49, 3651231326520682150499/343)
(86871401366965, 800240902996342804421)
(-14940295804037, -20312583017768395267)
(6964686011653, 2891018295500550053)
(-13822405039787, -29975099320903870267)
(9021833770315, -4686876167691073129)
(-9560836755947, 43269950494397842853)
(9269843446837, 6168182658272383877)
(473703694901557/49, -2890170815217234366301/343)

Commentary

New curve of rank ≥ 18 and log conductor 127.067, found as the specialization t = 17142/23 of the 2-parameter Mestre rank-12 family over Q(t) used by Fermigier in 'Une courbe elliptique définie sur Q de rang ≥ 22' (Acta Arith. 82 (1997)) — here at family parameters (u,v) = (2,5), sextuple {348, −600, −216, 492, 876, −900} — located by a discriminant-gated exhaustive specialization scan crossed with a staged Nagao-sum sieve; 18 independent points certified by positive-definite Néron–Tate height pairing. Conductor N (log N ≈ 127.07) is squarefree apart from 3², with bad primes 2, 3², 5, 13, 19, 23, 31, 37, 157, 163, 259627, 39576820317654329, 99173840155499332801.

last edited by cocoxhuang at 2026-06-25 03:15:38 · history