curve #5663

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y² + xy = x³ − x² − 52522475866099084386227311062117094x + 4654815207370493169734263092623504980282617955294384

a-invariants

[1, -1, 0, -52522475866099084386227311062117094, 4654815207370493169734263092623504980282617955294384]

c-invariants

[2521078841572756050538910930981620521, -4021760339168106087305548524949306075539082723957055445]

rank (lower bound)

≥ 24

torsion subgroup

trivial

conductor (N)

207305278122121647122102398619442186406224143695264547199350618664694926142867088683860319890176858

discriminant (Δ)

-87376272454580564558906451584514209379613789429542272940865494858507605640252477741005550789231083170988

Faltings height

18.7883

naive height

251.4626

primes of bad reduction

2, 13, 29, 43, 277427, 45934321, 3234901481815992014538352007, 11931089963207861994793682650012495295993252192041487

regulator

2530939265120226410347844242.0062408884694846293507304662649251781595198260254447380631

submitted by

apkorte

submitted at

2026-10-24 01:09:24 UTC

later contributions

primes of bad reduction recorded · Daniel Błażewicz · 2026-10-24 01:16:07 UTC

Witness: 24 independent points log in to add more points →

(159272191481830411, 18160494212114647803648285)
(122692523713095853, -7592530750827433369517343)
(-262719509212486458, -17893826703873510149169670)
(74272770486459633, -34110805120917424563411818)
(-357919481485959, -68363835313874960820941986)
(-63094440850893013, 87849406748312152480740445)
(570367787440817575/4, 64388146142006038914242251/8)
(290755168169492270, -118167928987187460045738312)
(-237490275689028118, -61102811388200258611726940)
(1143376460306557606/9, 154329165450037866666702046/27)
(-13002588540195397335/64, -42651997549615401780296138791/512)
(132832510464249190, -4677916635944982125279238)
(7901290328506735528/81, 15628688722195834953340057328/729)
(-22605759154033355/9, 1868025931272019637034601979/27)
(-34642411321970043, -80204402466420716133009167)
(3311303498200999629/25, -583419210860804682708601843/125)
(9529465040158966, -64460604153426634798589830)
(608848176881930488762/9, 15023150778394583795632998750214/27)
(759391898947017275447/6724, -7051518434694129286933269830805/551368)
(111331193196229405, -13687028821352496963975807)
(-25492270215523352, 77312133949432183906355520)
(121113554823023231587/4, -1332836025181670943939366018555/8)
(68602707502465410, -37074226120987998309722068)
(141319318952483757, 7394862479809865616094433)

Commentary

Specialization at T = -77/226 of the rank-17 fibration of X1008 given in curve#3594. Found by Claude (Anthropic): a Mestre-Nagao sieve over the fibration, then a 2-covering point search seeded with its generic sections.

last edited by apkorte at 2026-10-24 01:09:24 UTC · history