curve #6509

y² + xy + y = x³ − x² − 25443977028918972757097x + 1480011319201639338118091527889321

a-invariants: [1, -1, 1, -25443977028918972757097, 1480011319201639338118091527889321]
c-invariants: [1221310897388110692340641, -1278729779784720489095784581980840689]
rank (lower bound): ≥ 18
torsion subgroup: trivial
conductor (N): 62209683646987441959784598983560125347538651778630
discriminant (Δ): 107961725171300801101318183536361157187915956416016410364312640000000
Faltings height: 11.8074
naive height: 166.3859
primes of bad reduction: 2, 3, 5, 7, 271, 10037, 288348405564539, 125900273648686732933892417
regulator: 684494388065104.93579919480830319110777074028959403483343132598067696054248
submitted by: Michael Rubinstein
submitted at: 2026-11-04 04:55:19 UTC
last updated: 2026-11-04 04:55:19 UTC

Witness: 18 independent points log in to add more points →

(10431942550141/81, 13427996526674870626/729)
(329154499754871, -5971723015977112106276)
(5695314695569/49, 3305451863647336402/343)
(1289329516973449/1849, -45172398882623002443702/79507)
(20766012664929/289, -23425803671593990978/4913)
(-2127280339604609/11881, -22268531253176086377524/1295029)
(10001753106254511/22201, 944080664453086258706156/3307949)
(20303602246356639/24649, 2842210733359985768682892/3869893)
(8016668581786089/6889, 711343860202761785444618/571787)
(581459911179/4, 233655868187055737/8)
(-34107763887332121/187489, -762637070036227446691172/81182737)
(3748190260983441/33856, 27890741587708410415651/6229504)
(282811193133651/6241, 10105107807708409992016/493039)
(3247174160073711/22201, 98544712055249889016556/3307949)
(565953812803550031/2886601, 311265677242706293374916996/4904335099)
(-862135795991361/11449, -66750794839895353354108/1225043)
(250710888894448479/1841449, 58044271622093726055579532/2498846293)
(1246828240542249/9409, -19041497707729703540288/912673)

Commentary

Mestre ten-root construction, roots 0,18,70,86,102,109,149,157,166,195: 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 >= 18: the 18 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 04:55:19 UTC · history