curve #4124

y² + xy = x³ − 30084936999943557436114550250596774985704338964613506x + 2008497769652072620134170778694831238794155024896745604239521840153416838311620

a-invariants: [1, 0, 0, -30084936999943557436114550250596774985704338964613506, 2008497769652072620134170778694831238794155024896745604239521840153416838311620]
c-invariants: [1444076975997290756933498412028645199313808270301448289, -1735342072979390743795923554958449654314086076911035820105914668863264553753412113]
rank (lower bound): ≥ 20
torsion subgroup: ℤ/2ℤ
conductor (N): 122476517289985637066269292234016565764186430781473965549569412350148532718549910 ☆ record for ℤ/2ℤ torsion, rank ≥ 20
discriminant (Δ): 3365558868745136650666828961387914527384073636122108830582634654540238025686071069686474806834062026719261264833215143194693470999810839642095992816025600
Faltings height: 28.7449
naive height: 374.1212
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 23, 29, 31, 53, 67, 83, 89, 101, 107, 227, 241, 347, 509, 839, 1307, 1427, 1787, 2341, 2749, 10613, 21529, 25847, 27983, 47279, 51853
regulator: 297749506367161913856.906804155059234850132445483568735002607666178172776517630
submitted by: wgxli
submitted at: 2026-10-09 23:06:35 UTC
last updated: 2026-10-09 23:06:35 UTC

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

(100047648629730199390023028, 838899166587652886144771105771151646)
(100226099315487479886262708, 461831906439845934901861413541585006)
(99593646994734866728153396, 9383698324001186940572431576669178974)
(99623920911288782631958486, 8853820678983571416377895073850235010)
(99629931794782847176000708, 8748520033605305293699218357990997006)
(99971667934575290558950138, 2591695852728263787162049582177076356)
(99974915674677218826830176, 2527646392313393142821618081347186834)
(100009557996521527682104228, 1812522449270907817694341442137774126)
(100009722899548182371464888, 1808918226498857877672671375770239406)
(100241495228795035496157928, 1033365922632851979867072042190977646)
(100377501544575428418636388, 3848614122376665207537976228858738606)
(97486484539597312117849888, 45793103452123166691346355137807434706)
(99039119718076240567158148, 19020621677212219517113143480613754926)
(99121892959036349315118708, 17586868518590856835314811125114879006)
(99554462348736802382254276, 10068514680317575865831971335267487534)
(2298193630033463269870474, 1392612256672422918715647629507953130050)
(113263507415860566741984004, 232351071413903672186176126124463133390)
(142398697350111008394834628, 782250414675067477823108787767812089646)
(192744661606156525372228888, 1835850811334182970108337804428737496506)
(2473922979253264207306045036/25, 2555436022835096335676983189834976955926/125)

Commentary

specialization at T = 13149/1928 of the rank-9 fibration of X1152 given by: y^2 + (99239T^2 + 490032T + 695248)xy = x(x + (162T + 3601)(4805 - 671T)(9386T + 15495))(x - 23(49T - 883)(55T - 27)(188T + 39)(61T + 93)).

last edited by wgxli at 2026-10-09 23:06:35 UTC · history