curve #5541

y² = x³ + 210047643755937689748228x

a-invariants: [0, 0, 0, 210047643755937689748228, 0]
c-invariants: [-10082286900285009107914944, 0]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 58014480798713096047867061716102842721931808
discriminant (Δ): -593107500740673057928287900688637236692634518432664087309771540988694528
Faltings height: 12.4614
naive height: 172.7185
primes of bad reduction: 2, 3, 7, 13, 19, 23, 53, 113, 131, 139, 277, 307, 1217
regulator: 44091382285407525822.5612405127584824525443392493425909645813
submitted by: The Wren
submitted at: 2026-10-22 23:56:19 UTC
last updated: 2026-10-22 23:56:19 UTC

Witness: 11 independent points log in to add more points →

(78592819886382088831586575810076944/2329176576344708905378729, 300072802823052489392700189230151543706814950502896264/3554705456827670123842692043652328683)
(18370430004673148923288725885770896/33296445843768523057041, 3236887531455436446897576701716224589880337611026792/6075706934751213419924797108927161)
(198698943234867753738464713559056/506101332491184721225, -4305246594193318116987254040412055994044366420296/11385607839962651990435037842875)
(19607615891048711172428568890944/7186557361823515225, 88039904894812343187218762546226219888823826128/19265547012719475366409715875)
(1439217842108699477625725014379584/654525713416277422801, -55773016286132563096853895336528237213969366402416/16745189593161399427571834542951)
(516745393912577487513458657218234742318656/915254174367491245305111176281, 478443532071282355437167337340727195621578151787506132193948208/875613759462778976770418369374662836696841821)
(47181426739715012441276845812121/302221076401094131264, 1005095573280098486633573031158756899630114164803/5253964355380371841190347545088)
(22460016658898702861760664460219722293849360845307904/141984305399301485104764095369175313128025, 10316789781963980143879348984833035766194893013204971402185214680538303522913792/53500829032298263422459144386711098459075854553349338821260125)
(14694073715862836483256389526463447326262904689/62419008050062093459624182723216, 1781205979566511640079003378110740104766004771295319523494541185701239/493145749244961155350830168133569214670902700736)
(20903212374941229570646218449896138703768648070271729/65729236996583224745401174939206950194576, -5301197614391242179112729258279081121584875872176719370542382873025874156178999/16851473677889555555766091502026245879363013855194693369002176)
(5509425276897413794525075260376223545551801159166401951197209/60055319929039012809673231457294953766221095054400, -65886174941384552699117105292268866043874074848173308206910771020106613265857347529863703677/465400909169648501108297219486473541383983978718820929818240738255988928000)

Commentary

Rational 2-isogeny partner of my ICARM curve #5504: https://elliptic-rank.icarm.cloud/curve/5504 Curve #5504 was obtained from a curve from my own run at tuple (11,23,29,50), capture APH-07f1610bc9cb7d3079b579ba. Its model is y^2 = x^3 - 52511910938984422437057*x. Applying the rational 2-isogeny (x,y) -> (x + A/x, y*(1 - A/x^2)) with A = -52511910938984422437057 gives the submitted model y^2 = x^3 + 210047643755937689748228*x. The 11 submitted points are the exact transported images of the 11 witnesses on #5504; every image was checked exactly on the target. Rational isogeny preserves rank and conductor. The conductor is 58014480798713096047867061716102842721931808. This completes the rational 2-isogeny pair from the same own-run tuple capture.

last edited by The Wren at 2026-10-22 23:56:19 UTC · history