curve #449

y² + xy = x³ − 45985559764874647626604299762930x + 84640033039127748879277321178403309377568768900

a-invariants: [1, 0, 0, -45985559764874647626604299762930, 84640033039127748879277321178403309377568768900]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 11106328920975712642031651694736605793017517811417688175832290165042008870
discriminant (Δ): 3128819174937008750747092070091173185332512185826818190669176860413746229895787257077955072000000
Faltings height: 17.2218
naive height: 230.3312
primes of bad reduction: 2, 3, 5, 11, 13, 43, 61, 71, 51893, 7222702441, 37089253978986320869559544323368382227544182957607
regulator: 35580190380997350319.7316495500701916000885705473164554683237387284360685461456
submitted by: Bhavik Mehta
submitted at: 2026-08-30 15:55:02 UTC
last updated: 2026-08-30 15:55:02 UTC

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

(5734795874849160, -97609093234795677248130)
(1502746203307260, -137582625111736499564130)
(-1711663581811740, 397915849167082179687870)
(31654157020656900, -5508721937726902853547330)
(78841691012769540, 22057651178124792518681790)
(-7292367207508290, 179403460090673647099920)
(5682672382142160, 82633941050953526091870)
(1582581912395460, -125808577595529492291330)
(1809091016961540, -85841609034492870390210)
(33216639614275236, -5933512668799896612187458)
(2015294951188950, 12268382050821533353170)
(-3005449573404900, -442379821831059527404770)
(20008355951153700, -2678539895078416250054370)
(6421739058968820, 232716878971138779296910)
(5995117009297860, 156282713843277159567870)
(1529452986314820, 133734879463466125734270)
(-5533413661825788, 411911897195008870103742)
(2011543218432240, 16655682195695686551090)
(15576533162048500/9, -2731657001660230837855910/27)
(-6476418174195708, 332888743790694849258942)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = -3/7. The 3 points beyond the seventeen generic sections came from searching the 1311 norm-4 quartic covers of the Mordell-Weil lattice; the claimed lower bound is rank >= 20.

last edited by Bhavik Mehta at 2026-08-30 15:55:02 UTC · history