curve #410

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y² + xy + y = x³ − x² − 517097571914307467396942889454703x + 4517566384703220051910945882466217158874543213431

a-invariants: [1, -1, 1, -517097571914307467396942889454703, 4517566384703220051910945882466217158874543213431]
c-invariants: [24820683451886758435053258693825729, -3903177356383582013157981708960398667527941214188833]
rank (lower bound): ≥ 21
torsion subgroup: trivial
conductor (N): 91919939678480227622280532789556795313972539945305315872187122157174545321002490
discriminant (Δ): 32635308399185675469039987234791084282623661703291236419594278789894740368875979486423544707891200
Faltings height: 17.5956
naive height: 237.5910
primes of bad reduction: 2, 3, 5, 7, 13, 19, 43, 246811, 55659494882341145963786392282520892746547091175354264152150145565833
regulator: 4523743435808851842925.4387861538543746780952572787034340689824589284396516258574
submitted by: Andrii Prokhorenkov
submitted at: 2026-08-28 21:56:55 UTC
last updated: 2026-08-28 21:56:55 UTC

Witness: 21 independent points log in to add more points →

(4596298324570792106299130806689055219/3054834592353852261409, 326654371782498265821758389132208722798262040469135542758/168842412813418903395960965578673)
(10214359257173475, -549037469975506541476058)
(-99007296691336425525/22201, 8584695679669490406210369526958/3307949)
(3638256639246612386985823/267682321, 59481526439425520754822732287762730/4379550453881)
(-19747019283325728213/1369, 151749480803262863876836374478/50653)
(3280976685149882258844799/240157009, 203382695776482826271264412583515786/3721713168473)
(4934051460023269783/121, -9527177593872781428774587794/1331)
(10160592782039277188783871/819047161, -2555012185414563292171378378936215262/23440310700659)
(778838492076724425475/22801, -17793527524631140566439173778518/3442951)
(9648148221350658417407803/768232089, -1388868549068312892766264635229780082/21293088810813)
(-19138794873981960981101/734449, -343093889361374551270756046090682/629422793)
(-278038013622989192589/212521, -223235543835329905619346354671074/97972181)
(72976233071489205/16, -96084420546964826073465497/64)
(7754299934124268378843/727609, -289218608211798290055689607469778/620650477)
(4269394949290311052995/368449, 63869628162299715108546702922906/223648543)
(68465727580657049883295/5555449, -1688740805764603932321451281534294/13094193293)
(50603518854479908262192875/4086149929, 29866122420778101637836646037811234594/261198961911467)
(179596049404133871783699/20205025, 71716472610499035398319403495821178/90821587375)
(97055700525817393/9, 11932452349005060896575772/27)
(178041442239015480993565905/14234637481, 138029515003911576357382882779706962218/1698320363220629)
(182687379793222367278936465/14764437081, 209886054787704646607638926491652605428/1794011985275229)

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