curve #6095

y² = x³ − 143352735666059647x + 20308350490107193886372290

a-invariants: [0, 0, 0, -143352735666059647, 20308350490107193886372290]
c-invariants: [6880931311970863056, -17546414823452615517825658560]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 837877902601733961103246999529573248471046104
discriminant (Δ): 10368209505862013472789264367074766259536161835862272
Faltings height: 8.7554
naive height: 130.1259
primes of bad reduction: 2, 17, 23, 43, 4447, 8822239, 9340081529363221267876573591
regulator: 102081371964861448757.346950941231692484623413216399223990417493970261364
submitted by: Michael Rubinstein
submitted at: 2026-10-29 02:21:57 UTC
last updated: 2026-10-29 02:21:57 UTC

Witness: 17 independent points log in to add more points →

(3787234091, 231944464397728)
(-11994361725034/54289, 81188295259488254108/12649337)
(-63717868777/169, 10010594727765256/2197)
(441472627/9, 98816186234152/27)
(37085072170/121, 3024154650391010/1331)
(50577646321/324, 7680634860145103/5832)
(9845643444019/52441, 1325579837889527504/12008989)
(27821079492921/151321, 24057500199652660778/58863869)
(65895007605373/863041, 2510967927667103806214/801765089)
(73867808561066/151321, 480575559881387604398/58863869)
(77493813526519/447561, 246576925306805895064/299418309)
(87945272342803/241081, 481688524238209745504/118370771)
(163009976472017/654481, 123780103901550255358/529475129)
(200638496008633/274576, 2515717415395246704301/143877824)
(2071876749637501/12047841, 36015112609708608022778/41818056111)
(4157825023170998/22118209, 6047942239649960436796/104021936927)
(33448815372866853/40998409, 5546448119454114002708254/262512812827)

Commentary

Mestre ten-root construction, roots 0,19,23,31,37,51,57,81,93,112: with p(x) = prod (x - a_i), write p = q^2 - r with q a monic quintic and deg r <= 4; the genus-one curve y^2 = r(x) carries the ten points (a_i, q(a_i)), and the curve submitted is the minimal Weierstrass model of its Jacobian. Found by enumerating integer ten-root sets 0 = a_1 < ... < a_10 and scoring each by S = sum of -a_p log(p)/p over the good primes 5 <= p <= 199 (with early abort on low partial sums); the sets with S >= 31, or S >= 30 and one bad prime in that range, or S >= 29 and two or more bad primes, were screened by exact conductor and by a GRH explicit-formula upper bound on the rank, and the survivors were certified with ellrank. Rank 17, proved: 17 independent points listed, matching the 2-descent upper bound. Torsion trivial.

last edited by Michael Rubinstein at 2026-10-29 02:21:58 UTC · history