curve #216

y² + xy = x³ − 292164008972198360516541x + 60777760375298123458072491954931825

a-invariants: [1, 0, 0, -292164008972198360516541, 60777760375298123458072491954931825]
rank (lower bound): ≥ 17
conductor (N): 8229621766982592642433268362527136439092061141374442470
naive height: 173.7084
Faltings height: 12.1714
discriminant (Δ): 319660562013917792358156721464086339946753006672777251625449427814400
primes of bad reduction: 2, 5, 11, 29, 37, 41, 61, 257, 2467, 4177, 40298971, 453713587, 575746473701180535499
regulator: 605706094414059953.433928438056900702744283818617694394403423434391605716
submitted by: Edgar Costa
last updated: 2026-07-01 22:35:41

Witness: 17 independent points

(-38718845802698642844526436/105831648077401, -375098011016010771236367933056744517569/1088737893845506994851)
(370022648858, 57729896660684391)
(813867360018, -601735013298569809)
(51388025011062/169, -16041282810910889293/2197)
(1742212142545418/5041, 11787551290092658020681/357911)
(138421495063902/49, -1600711642945571193647/343)
(1523744473018, 1775786428781568391)
(88651233252202/289, -22108110499592240017/4913)
(72435267588973433/18496, -19318466609689318953185999/2515456)
(308271320498, 2714843230760431)
(82306269319150/729, 3365202373896362951845/19683)
(2782325464450/9, 37096591825050565/27)
(5790971164122842/18769, 6054012515438553117423/2571353)
(-5029167919947/16, -22313592130959039811/64)
(1262165031167/4, 18298327651158693/8)
(556113986993050/1849, 841917866240162077525/79507)
(2466561849275650/289, 122260608489486622716095/4913)

Commentary

Specialization at T = -2407/2 of the Mestre/Fermigier family y^2 = r(x,T), where r is the degree-<=4 remainder in p6(x-T)*p6(x+T) = g(x)^2 - r (g monic of degree 6), p6(x) = prod_i(x-a_i) on the sextuple (a) = (-44, -60, -6, 110, 94, -94). Located by a Mestre-Nagao sieve of this family for small-conductor high-rank specializations. Seventeen independent rational points were found by rational-x enumeration on the quartic plus a direct minimal-model x = n/q^2 sieve; rank >= 17 follows from the positive-definiteness of their 17x17 Neron-Tate height-pairing matrix, computed independently in Sage and Magma. Conductor by Tate's algorithm. Proven lower bound on the Mordell-Weil rank (no exact-rank, Selmer, or BSD claim).

last edited by Edgar Costa at 2026-06-25 21:52:44 · history