curve #4298

y² = x³ + x² − 545476165159141642291931774x + 4896262650962524332198809518887171713256

a-invariants: [0, 1, 0, -545476165159141642291931774, 4896262650962524332198809518887171713256]
c-invariants: [26182855927638798830012725168, -4230370930431778120155337257111496436604160]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 81227180467392566378365208705376
discriminant (Δ): 30907266709720414267190667657062061801689969180627815432267126636683017109332544
Faltings height: 14.1469
naive height: 196.3047
primes of bad reduction: 2, 3, 11, 13, 17, 31, 43, 89, 97, 167, 683, 733, 1031, 12611, 27823
regulator: 842224904703867.7964698149276723567353180606186645036437862
submitted by: The Wren
submitted at: 2026-10-16 19:04:50 UTC
last updated: 2026-10-16 19:04:50 UTC

Witness: 10 independent points log in to add more points →

(48549460337355722374/1755625, 243522431625935448473734315068/2326203125)
(2420772274936095682606/15062161, -117911906298516955408759368046368/58456246841)
(16459190120717334214/243049, -63236168590652891886217048884/119823157)
(38205792468637311698134/2739161569, -175465561894271781341402229906636/143359499036753)
(41787055350251809780100100349/2973950830059364, -393120358601791887928174479648947840982141/162181262077309212932888)
(7325724437040111834824086/504305921025, 2200694521640652075390984859359455884/358130328286298625)
(27628830325427408582431591297777/1408749964281383824, 69785912759049123580189085533297248352375498545/1672056225060298284365327168)
(84955933716754684845340366/5941898635201, 64789637056327572668109341635015924032/14483978055064592801)
(45571547602955774589000279018837073/1077558807427585647376, 8479617968235286422358673668473311427606418399808385/35372151092732853008602604541376)
(7820667691245255193421395450060077841/53074828564999201960000, -21611323032394510234549471360505763121338158189794862361/12227365661792614906745427256000000)

Commentary

Obtained by a rational 2-isogeny from curve#4251, submitted by Michael Rubinstein. The source is a specialization at t = -167/1118 of the Elkies-Klagsbrun Z/4 fibration E2. This isogenous curve has the same rank and conductor, with torsion subgroup Z/2 x Z/2. The ten supplied points were transported from the source curve.

last edited by The Wren at 2026-10-16 19:04:50 UTC · history