curve #199

y² + xy = x³ − 20820207864197471248300179976626x + 36732936589138673862895758597955508398047757956

a-invariants: [1, 0, 0, -20820207864197471248300179976626, 36732936589138673862895758597955508398047757956]
rank (lower bound): ≥ 13
conductor (N): 21785392458764315483988614758901932764423833726404496768609056369259382575196330
naive height: 227.9632
Faltings height: 16.8288
discriminant (Δ): -5290724783356970045570091261064369041862691109828850503901233859803837652758778996437693235200
primes of bad reduction: 2, 3, 5, 7, 13, 19, 86044359449701746413144681, 4881201479084572277272794646126863424004819546339
regulator: 3144017645707793.395706961347052974878979749568459122697271441584
submitted by: Seewoo Lee
last updated: 2026-07-01 22:46:36

Witness: 13 independent points

(1162634353995497656800804/7778521, 1253039186201942747498661935105222982/21694295069)
(3627637297190869531/36, 6902364641611559233412383477/216)
(1558179052657862437/144, 1794466997628157186816731475/1728)
(3881968959990696, 120039898039176624614682)
(2837185351207296, 22369929495303810169122)
(133369750392561176/49, 5181060484570817333704274/343)
(290049012608047521903204/131859289, 60156526711202358713019503533166394/1514140215587)
(27074001988877942159/6724, -74342839994965136661541523599/551368)
(2480181137075511944/289, -3439495665410956315384662434/4913)
(9363687175740228, -814109553204195618921090)
(4354296816133898977943076/338376025, -8579048965788332093253582643648116486/6224426979875)
(11397032844750127922575524/4282000969, 3685010112080220968145792764888880246/280201297408453)
(19369629507101316, -2626899353324391553849218)

Commentary

Rank ≥13. Mestre–Fermigier construction from the integer 6-tuple a=[240, -1692, -996, 1260, 1776, -588] with shift t=2789/9: 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 13 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 16:10:19 · history