curve #213

y² + xy = x³ − 74540773450198004669118078904466x + 244066692939635107384386902977544619850388014596

a-invariants: [1, 0, 0, -74540773450198004669118078904466, 244066692939635107384386902977544619850388014596]
rank (lower bound): ≥ 16
torsion subgroup: trivial
conductor (N): 10465426762586441668769337684707368604491256699170701827399830238043087368190
naive height: 231.7803
Faltings height: 17.1972
discriminant (Δ): 773452069750637843410919255151077621482402538193851004785076611405015441616047124089938250956800
primes of bad reduction: 2, 3, 5, 7, 11, 17, 19, 157, 6651361, 21871169, 117189223, 2264709407, 136116631176129536978806192189908923
regulator: 3320587594876995144.037797883037040736349990143118593145621928048769962
submitted by: Seewoo Lee
submitted at: 2026-06-25 16:04:49 UTC
last updated: 2026-08-24 04:58:52 UTC

Witness: 16 independent points

(11421650765628186754/9, -38599643258316838007145121946/27)
(5657778683304010828/81, -13359296359452931217293552370/729)
(1859392181426721484548004/45873529, -2482023862685444193093304134783782266/310701411917)
(6670385051890276, -208910006765036920615298)
(953233885740561876/169, -121770304938003450603963594/2197)
(787993857911111269/144, -9344055086171434479140719/1728)
(5476037191636420, 9406440863363801143966)
(154182032082462285/16, 41506127422013923347712539/64)
(17650888545826756, 2104176405239656414694302)
(22770019726862068, 3217519285772321257814830)
(371978685111016429303396/10569001, 220592085961447243696290765616630922/34359822251)
(10776150844881796, -831979087315908337590818)
(473333385712074805507238176/80111075521, 2257157791353302661000762308145736306498/22674558704388319)
(125469021849327376, 44340551401859153192147902)
(687807339571871371/100, -238237168480389111912179261/1000)
(12933741469020282429244/1646089, 799917654142229025008395406761034/2111932187)

Commentary

Rank ≥12. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=6773/18: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 12 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.

last edited by Seewoo Lee at 2026-06-25 16:11:26 UTC · history