curve #212

y² = x³ + x² − 221556180740323405132844117936x + 35386140191724122461245294467670188433973860

a-invariants: [0, 1, 0, -221556180740323405132844117936, 35386140191724122461245294467670188433973860]
rank (lower bound): ≥ 11
conductor (N): 1103561624055499058867562340698878392772504928025988266715523317532246643920
naive height: 214.3250
Faltings height: 15.8427
discriminant (Δ): 155094517685876658621636759147912906994801674312562010974276152381499783783841112416793600
primes of bad reduction: 2, 3, 5, 11, 31, 157, 670606297099, 1575838430456954508271967, 81274068710384465721193186106423
regulator: 15079211501874.5923514849165940580973990610604958433553929196
submitted by: Seewoo Lee
last updated: 2026-07-01 22:49:33

Witness: 11 independent points

(1439725582641991678/9, -1727498601505665033867479240/27)
(1609812018760123810/81, -2041934892838682216734870460/729)
(382889599831597753/144, -233409324592028354395076875/1728)
(1461300830275451952/841, -1706610970573605828131905230/24389)
(3003181201931493/4, -136765210570118860311615/8)
(4642447936168800893203/13344409, 31359710078558692581991890119040/48747126077)
(24321114180952583584546/63696361, 1291845898306026112352226583813164/508360657141)
(38661152893172916450454/78234025, 4722711530104732719979077042193992/691979951125)
(721344004004635, 15840139419502659908460)
(779580898158616, 18342645394695663676806)
(16344491930375377/16, -1893202461186603241978305/64)

Commentary

Rank ≥11. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=14612/9: 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 16:11:22 · history