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curve #211

y2 + xy = x3 − 2164487921238768499500640803351x + 1226101128097099442256424377019056466085320681
a-invariants
[1, 0, 0, -2164487921238768499500640803351, 1226101128097099442256424377019056466085320681]
rank (lower bound)
≥ 11
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
6574796646890.56238082668096592613712251120723802057237124620
submitted by
Seewoo Lee
last updated
2026-07-01 22:49:33

Witness: 11 independent points

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 · history

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