curve #6517

y² = x³ − 15384564398667087x + 761781563325326731371570

a-invariants: [0, 0, 0, -15384564398667087, 761781563325326731371570]
c-invariants: [738459091136020176, -658179270713082295905036480]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 6810058810005453132518759329541962443430063224
discriminant (Δ): -17651672435534134519488624182172766653370723876608
Faltings height: 8.2148
naive height: 123.5030
primes of bad reduction: 2, 3, 1721308271520361, 54948989500539799425066454847
regulator: 155866337221101.604160058433357152740957920067288092827540300692270609670
submitted by: Michael Rubinstein
submitted at: 2026-11-04 04:56:15 UTC
last updated: 2026-11-04 04:56:15 UTC

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

(-572475543/4, 1433249256573/8)
(441718858437657/256, -9283653242106855044733/4096)
(-945021017913/6889, -308344365203381928/571787)
(11649879946101/11449, -39481239778078976838/1225043)
(6233082346417/15129, -14928251088894178598/1860867)
(5697977627721/85264, -4461760918817580357/24897088)
(2218610620854/22801, 1485156041809957656/3442951)
(60504065433/484, 9476723282597829/10648)
(5654586825747/26569, 11561989999879805832/4330747)
(-11014675703399/83521, 17017060699180670498/24137569)
(-1493616820703/11881, 1090463417468081558/1295029)
(200396397033/3844, -75904043026151619/238328)
(-320085924671/2500, 99427483812147767/125000)
(830119369113/33856, -3936465662253039549/6229504)
(20247285, 677190024630)
(-3365429943072663/35081929, 241860517208570847206898/207790265467)
(21838554764833/401956, -74864919672557906599/254840104)

Commentary

Mestre ten-root construction, roots 0,8,83,107,123,146,151,154,163,183: 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 >= 17: the 17 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 04:56:15 UTC · history