curve #211

y² + xy = x³ − 2164487921238768499500640803351x + 1226101128097099442256424377019056466085320681

a-invariants: [1, 0, 0, -2164487921238768499500640803351, 1226101128097099442256424377019056466085320681]
rank (lower bound): ≥ 15
torsion subgroup: trivial
conductor (N): 76884160347970864077673900010863037738725767115232422182209303915570
naive height: 221.1635
Faltings height: 16.1652
discriminant (Δ): -434796625061430848546124049916980604145395749660512866802185034140618458039551190139699200
primes of bad reduction: 2, 5, 7, 13, 17, 23, 31, 157, 33196799, 2096948303, 1944251946437, 25233503438885314304619774163
regulator: 12148241829565057.484009176480523989881551913829824169221494147482977
submitted by: Seewoo Lee
submitted at: 2026-06-25 16:04:39 UTC
last updated: 2026-08-24 04:59:05 UTC

Witness: 15 independent points

(193471633301052616639334/1352569, -85094808570897910903821838655812881/1573037747)
(405207134144905894/9, -257802161495288662231116521/27)
(550091167141970389100/130321, -12204497223776414581241469614979/47045881)
(209300708887435110/361, 88389056748088433517564679/6859)
(793448046452518, 2866253390847396035317)
(846049037447238, 662750304475737010357)
(7900678100479420/9, 42649563397126798370693/27)
(1030036895154710, 9457437854839529239509)
(1710543209524774, 50285471431872021894373)
(2517886218947724, 108346380885156670615183)
(51824229593028927046/15625, 341289027873148644301074534281/1953125)
(277588490992421308450/33489, -4556521484245793418116285107717/6128487)
(3111628798699919/4, -28925121330387869957959/8)
(16599521293635094/9, -1599200283280064725873321/27)
(357853175069383802/289, -103579034263120548599051955/4913)

Commentary

Rank ≥11. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=2326/23: 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 16:11:17 UTC · history