curve #758

y² + xy = x³ − 4659208505549796192566877972738x + 3872339028913665740859155632055568027325930692

a-invariants: [1, 0, 0, -4659208505549796192566877972738, 3872339028913665740859155632055568027325930692]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 23789261559055379338105234248315974632795330743806015189919339103950
discriminant (Δ): -4683076378200201054706808777044542971578135645187890483155129326284455211358884882821120000
Faltings height: 16.3595
naive height: 223.4635
primes of bad reduction: 2, 3, 5, 13, 17, 19, 37, 43, 89, 1051, 154609219, 707424629, 644635416755471, 3599574785289150212183
regulator: 93675267828215863394.6173115996625124458830483395767028481609434647588793529193
submitted by: Valery Asiryan
submitted at: 2026-09-18 02:14:59 UTC
last updated: 2026-09-18 02:14:59 UTC

Witness: 20 independent points log in to add more points →

(-245908406229948, -70733353065492736071666)
(58011663190477176/121, -55659874201641240867863766/1331)
(731292312063258, -29260594129503168993150)
(1689332227554852, -28678869920446921176786)
(788286571512899976/625, -23430593672510390284818726/15625)
(1276594700307222, -2208371861138495771646)
(474704254692502272/361, -30206866539963857400281574/6859)
(-1191510361576668, -87933216243133624529106)
(61170591518809938/49, -408356460720165166325388/343)
(1378002767490852, -8283401645920574826066)
(3569640265334309292/361, -6595472578598442958043906334/6859)
(2765176858041192, -110144938785972987955986)
(4961997927393372, -320818967190758419007346)
(1665296047336932, -27048008396916245539026)
(438315503443445862972/326041, -1152900711987926313995881154046/186169411)
(3111265842932772, -139618334094080356686546)
(88623450578371767051935518307748/76975946096924449, -124934608766501342877095240680543833070918102962/21356621616408019844113457)
(91113527945731327088505199900565478318/75058087492735178646049, -47208398897612300480064259024690292721752420382578364432/20563462398661982316313519053716657)
(179582859760718634822/130321, -389703570061702450537962908136/47045881)
(75400529481812854437984/21836929, -17362934646071823485113248983641842/102043969217)

Commentary

Specialization at T = -2/9 of the rank-16 fibration of X948 described by wgxli for ICARM #625. Starting from 4 elementary polynomial sections, an independent rational-point search yields 20 points with an exact independence certificate, proving rank >= 20.

last edited by Valery Asiryan at 2026-09-18 02:14:59 UTC · history