curve #94

y² + xy = x³ − 52751556365588628145400492670x + 4655848829189654008696988351827664455270212

a-invariants: [1, 0, 0, -52751556365588628145400492670, 4655848829189654008696988351827664455270212]
rank (lower bound): ≥ 18
conductor (N): 42682665107572966519161676237698352985292192192557605656213390
naive height: 210.0198
Faltings height: 15.2929
discriminant (Δ): 30329481606266546964270732939527654191747398716945306980836010472365688288267827200000
primes of bad reduction: 2, 3, 5, 11, 13, 19, 31, 43, 47, 79, 9993701, 10586665361105868350008239735615374993841941
regulator: 65246279290282792.166077869689225539534770704132049918615908028178898997037
submitted by: David Renshaw
last updated: 2026-07-01 22:33:40

Witness: 18 independent points

(200212896884244, -1455981468425261901102)
(128196315395964, -10477319710847678382)
(-205036993618356, -2617647609254374848702)
(105878869571004, 507455632507918359858)
(144206806357572, -218135007877172188782)
(781870376871564, -21009164439672659945982)
(126484045207764, -84483037515959599782)
(-9476651858696364/49, -947083054999256026387626/343)
(18548018008764, -1919321000484069899982)
(141801409967604, 163998231164818681938)
(592900127539964, 13483395378366317086418)
(196304908050924, 1365719545267889883378)
(19275260524764, 1909505236219745647218)
(128085575458524, 22279824368584248498)
(7388129939825436/49, 123606160234861151139774/343)
(124886954998764, -125293276775632768782)
(127624354397784, 46988285799809697078)
(1001741371622664, -30936154910786659336782)

Commentary

New curve of rank >= 18, found as the specialization T = 1315/12 of the 2-parameter Mestre rank-12 family over Q(t) used by Fermigier (Acta Arith. 82 (1997), sextuple {0,55,314,378,1007,1036}), located by a staged Nagao-sum sieve; submitted for its small conductor at this rank. 18 independent points certified by positive-definite Neron-Tate height pairing.

last edited by David Renshaw at 2026-06-12 02:09:41 · history