curve #6175

y² + xy = x³ − 20041798012414773644650x + 1091040202112161863735130035418532

a-invariants: [1, 0, 0, -20041798012414773644650, 1091040202112161863735130035418532]
c-invariants: [962006304595909134943201, -942658734626350859724046214304026449]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 4111423344469783039462650315957622792007824170
discriminant (Δ): 977512615005441240448081346977350672230303316160000170424320000000
Faltings height: 11.5774
naive height: 165.6699
primes of bad reduction: 2, 3, 5, 7, 17, 1151659200131591887804663954049754283475581
regulator: 503112787770038352205.914050451455772934315842224602834646652385911571998
submitted by: Michael Rubinstein
submitted at: 2026-10-30 14:40:11 UTC
last updated: 2026-10-30 14:40:11 UTC

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

(86912368454, 2382229174010048)
(89068018804, 3541639574358598)
(-260368456797376556/1907161, 94327824796376019426055918/2633789341)
(-3202343404488296/19881, 33183062327308914577598/2803221)
(36709256155924/841, 421999477512282878942/24389)
(1227025726597364/14641, 284009273083410703478/1771561)
(1769818078512212/52441, 255623109454659054360302/12008989)
(3300202899464684/3481, 187585038784538133115762/205379)
(11704096499525924/3481, 1265108162349029153463442/205379)
(24796523094621524/502681, 5315490150974115903085742/356400829)
(100323595040580724/1985281, 40274621514272410860451142/2797260929)
(329675891238309956/4338889, 23722157313774398512985426/9037905787)
(1163865422333886644/1907161, 1224449167439307035889439118/2633789341)
(9904451274376521886/115627009, 2080058293416924976106894746/1243337227777)
(10817821672974115724/129026881, 362245046690878226591443442/1465616341279)
(13813295670219982196/179372449, 5001409156411887626632746086/2402335209457)
(711604499283390482834/10520399761, 7228476136182485085634549388692/1079066883086009)

Commentary

Mestre ten-root construction, roots 0,9,31,43,80,81,83,104,110,129: 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); the sets with S >= 31, or S >= 30 and one bad prime in that range, or S >= 29 and two or more bad primes, were screened by exact conductor and by a GRH explicit-formula upper bound on the rank, and the survivors were certified with ellrank. Rank 17, proved: 17 independent points listed, matching the 2-descent upper bound. Torsion trivial.

last edited by Michael Rubinstein at 2026-10-30 14:40:11 UTC · history