curve #6508

y² + xy + y = x³ − x² − 114546485377913284632x + 470355099486874881428017771035

a-invariants: [1, -1, 1, -114546485377913284632, 470355099486874881428017771035]
c-invariants: [5498231298139837662321, -406386805931917856712178084694025]
rank (lower bound): ≥ 18
torsion subgroup: trivial
conductor (N): 58355173931578483333998489018706959609875067416042
discriminant (Δ): 615917755882186101189757544362919497048390990719941354586112
Faltings height: 10.3325
naive height: 150.1761
primes of bad reduction: 2, 11, 274962639936119, 88016446198598609, 109602174072609641
regulator: 1933466697304407.6050319846549472013884767410311885133109484085654778908258
submitted by: Michael Rubinstein
submitted at: 2026-11-04 04:55:12 UTC
last updated: 2026-11-04 04:55:12 UTC

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

(43917729901, -8952496567877303)
(40348504813, -7844509691391671)
(-12211713557, -219266799220173)
(443069501395/81, 62934396369837011/729)
(472187536957/81, -19440281779475615/729)
(6758134900301/1225, -3410842457819398181/42875)
(101167570432901/15625, 22427942517021617149/1953125)
(1346738780633/169, 554032673800003445/2197)
(232488208016333/28561, 1342013571702192291553/4826809)
(6484774957, -15708721449015)
(8203317100609/1225, 2568125397970693707/42875)
(6372922570634203/956484, -50698551203457604414571/935441352)
(-70194142691/9, 25461977314215019/27)
(-223389226764571/24025, -3185020083793529988293/3723875)
(-544283997171/2209, -73308413498812092089/103823)
(2488479684553908037/382710969, -159799606131125445021966037/7486974686547)
(6965888269, -102214306077335)
(2514595437333/529, -2219226779831861625/12167)

Commentary

Mestre ten-root construction, roots 0,1,4,12,33,35,125,143,169,184: 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 04:55:12 UTC · history