curve #210

y² + xy + y = x³ + x² − 14678039879640223236646547506x + 596149475496280105667226784565254703843519

a-invariants: [1, 1, 1, -14678039879640223236646547506, 596149475496280105667226784565254703843519]
rank (lower bound): ≥ 15
torsion subgroup: trivial
conductor (N): 44702660536826547810138278991954523939494764170080309889431835489890
naive height: 206.1821
Faltings height: 15.1685
discriminant (Δ): 48857429671295622571237789586614535277793367429869518495768693667778413208750462668800
primes of bad reduction: 2, 3, 5, 7, 13, 17, 19, 157, 2004720611, 630581423473, 255430951802014070304557438075659110521
regulator: 4263827675457975.8484504916783373274615394841523187783167840060691629
submitted by: Seewoo Lee
submitted at: 2026-06-25 16:04:34 UTC
last updated: 2026-08-24 04:59:00 UTC

Witness: 15 independent points

(58446067465881769, -14129669742311093189766125)
(59805178857601951/9, -14622998659211562714239875/27)
(41268971140291161/64, -8243875117670686839240415/512)
(308108574052911809/841, -162496149169433149662813345/24389)
(2136555534033197/16, -64549605615325939384071/64)
(2447918870143671799281/27321529, 2428644462538774240003361334025/142809632083)
(91033697994269, 119840245127165271375)
(3579092917301768024617/34539129, 87981230877416815421786034759967/202986461133)
(8040116944201663249569/45010681, 578803785815401822474435058544295/301976658829)
(324903547043833, 5488603739649485397427)
(360854332952117, 6502958279889840406551)
(-53214619278427, 1107493976537428877427)
(6482180240715071/25, -471403564093113047225109/125)
(653921430673935762685701/5853321049, -265195372417430981976864914121869735/447820033495843)
(1203469842185308462241/7198489, -32404587928119090182807522993175/19313545987)

Commentary

Rank ≥11. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=15681/13: 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 11 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 16:11:12 UTC · history