curve #175

y² + xy = x³ − 29126329596087483849125163x + 62423713413815642823213203923492112817

a-invariants: [1, 0, 0, -29126329596087483849125163, 62423713413815642823213203923492112817]
rank (lower bound): ≥ 16
conductor (N): 2199428785729749633481682887731454385883881335809165450
naive height: 187.5772
Faltings height: 13.5472
discriminant (Δ): -101999356220239409114706029907182154313690823961385846723890824782205184000000
primes of bad reduction: 2, 3, 5, 11, 17, 19, 31, 43, 73, 8788437716942464397, 4825682552870583764479687
regulator: 393519862286889.6501035029965339047136645394249099052522877344608375237
submitted by: Seewoo Lee
last updated: 2026-07-01 22:40:38

Witness: 16 independent points

(5110339735435422, -365320817187192230444511)
(497618622929982, -11099911453301598239391)
(138299206337886, -1625188557246978877791)
(-1172961026178, 9745458965588806689)
(1451827418232, 4816375036731350409)
(1858357791822, 3835934257381893489)
(2360920940922, 2611205774465224089)
(2902124502102, 1529089488635552049)
(30210396731848/9, 42492257648703299603/27)
(1537958119559502/361, 27143958826181392261131/6859)
(860963638869174/169, 14942816278015024293285/2197)
(5129428439172, 6926964084183165339)
(11766270056735352/529, 1241901508300659121973055/12167)
(34194892011582, 197611265911196429409)
(185624555321995496598/5257849, 2501114969242211280352058304093/12056247757)
(107346062118846/25, 508148539017655964541/125)

Commentary

Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-1146, -2304, -654, 3054, 2880, -1830] with shift t=1907/1: 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 16 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 16:08:24 · history