curve #6486

y² + xy = x³ − x² − 2246946621357532095234x + 41545546795457030407521576974544

a-invariants: [1, -1, 0, -2246946621357532095234, 41545546795457030407521576974544]
c-invariants: [107853437825161540571241, -35895352430789533801885415573435445]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 2492992669762339391373056866125452214880825163390514171138798
discriminant (Δ): -19609880340350561652540465308942807122252570735229784470177785068
Faltings height: 11.1383
naive height: 159.1318
primes of bad reduction: 2, 3, 19, 23, 467418881, 41235613974633777513431, 5481091952843413380510791
regulator: 2103702594994226935.73476742595392354959001891784813913535252583245498714350503
submitted by: Michael Rubinstein
submitted at: 2026-11-04 04:52:34 UTC
last updated: 2026-11-04 04:52:34 UTC

Witness: 20 independent points log in to add more points →

(533051318728/9, -291090331253911624/27)
(15317673335478, -59949723413749857780)
(-5433186562628823/153664, 527849597227259830085151/60236288)
(38469901900497/4096, 1209007313875919261817/262144)
(181943398799823/7396, 684304184080735056237/636056)
(29278279749097/1024, 26919073795094178811/32768)
(327845545096461/10000, 1765805696267956031241/1000000)
(7001074074049546/82369, 509553508661859248512234/23639903)
(398772279729997/2704, 7594891053397509069291/140608)
(-2295023571090/49, -2276186841414390414/343)
(39685439322, -3857017953083952)
(25458081546, -917799961267296)
(213649800308084655/4553956, -60994002406829837765611263/9718142104)
(398476877943/121, -7781700649676884413/1331)
(-177071009456942/5929, -4134412260787016144280/456533)
(404891128690250941/2839225, -244936283766915862788552266/4784094125)
(8916313174322049997/184226329, -16990056044680801424598076666/2500503963517)
(621487116833746/11025, -11224242753302190317096/1157625)
(7761678087762/49, -20751074404917855270/343)
(907687748860665135/11978521, -725677702627828302240462243/41457661181)

Commentary

Mestre ten-root construction, roots 0,9,56,64,86,96,100,123,156,180: 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); for the sets with S >= 31, or S >= 30 and one bad prime in that range, or S >= 29 and two or more bad primes, rational points were found by a point search on the quartic y^2 = r(x), and the curves were screened by conductor and by a GRH explicit-formula upper bound on the rank. Rank >= 20: the 20 listed points come from the point search and are independent (height matrix nonsingular at 57 and 115 digits). Torsion trivial.

last edited by Michael Rubinstein at 2026-11-04 04:52:34 UTC · history