curve #4173

y² = x³ + x² − 19406500882447470905x + 35342204887178656570893130903

a-invariants: [0, 1, 0, -19406500882447470905, 35342204887178656570893130903]
c-invariants: [931512042357478603456, -30535665028111431531396536720896]
rank (lower bound): ≥ 16
torsion subgroup: trivial
conductor (N): 13767280447195777685103339403832411097298898560
discriminant (Δ): -71840370513891307772451042284388553024762942729977472000000
Faltings height: 10.0371
naive height: 144.9929
primes of bad reduction: 2, 5, 7, 29, 43, 61, 330907, 136789181, 892516293545548488157723
regulator: 143003248156445.7808323816230459406538076024216663213193179486649309246
submitted by: Andrii Prokhorenkov
submitted at: 2026-10-11 22:58:08 UTC
last updated: 2026-10-11 22:58:08 UTC

Witness: 16 independent points log in to add more points →

(13994577826, 1582583146150825)
(2959307794959/169, 4944338486911214860/2197)
(622576157598939/1849, 15532871707026579973340/79507)
(-343181002156931/131044, -12388830225238632562075/47437928)
(-539228522214/121, 242965243114661305/1331)
(133963761779/625, 2759590154664084708/15625)
(210562915230799/94249, 1620569019612979186100/28934443)
(15350074834/9, 2291753724140125/27)
(12987201262209/4624, 17199794611192128215/314432)
(1745366481596594/375769, 49085910902437545724875/230346397)
(266798863679/169, -204212862263347140/2197)
(12835323089469/1369, -41694220779970453100/50653)
(12525943711977946/5013121, -555789773290698916957925/11224377919)
(42590753203, -8744555419896268)
(27596548518652279/528529, -4568621735612671022370700/384240583)
(27630300786539/4489, -116128012003624280100/300763)

Commentary

Mestre construction: sextuple [-650, -481, 74, 225, 256, 576], T = 23; y^2 = r(x) with p(x-T)p(x+T) = g(x)^2 - r(x), p(x) = prod(x - a_i), on the locus sum a_i = 0, 12 p5 = 5 p2 p3. Found by a discriminant-first sweep + Nagao-sum rescoring + ratpoints search on the quartic.

last edited by Andrii Prokhorenkov at 2026-10-11 22:58:08 UTC · history