curve #3237

y² + xy = x³ − 156948296090453820673562578252548774710851738x + 756767158289903494103996663814727483895600061497503186459992789092

a-invariants: [1, 0, 0, -156948296090453820673562578252548774710851738, 756767158289903494103996663814727483895600061497503186459992789092]
rank (lower bound): ≥ 18
torsion subgroup: ℤ/2ℤ
conductor (N): 235186508255669484903975893032399896944786213691049874204261617969438950
discriminant (Δ): 23636472798687452891336930281743911668145286194574755817126493710872140632724179381635527787273374641306443340460123043587584000000
Faltings height: 24.0820
naive height: 316.9071
primes of bad reduction: 2, 3, 5, 11, 13, 31, 37, 47, 71, 73, 79, 137, 149, 163, 307, 601, 1063, 1187, 1229, 3313, 5261, 7229, 7879, 15131, 59999, 363833
regulator: 12901794600198769348.516381751324979263940895866339250512755555249216913915
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:44:04 UTC
last updated: 2026-09-26 20:44:04 UTC

Witness: 18 independent points log in to add more points →

(25870722056910014906828, 3743197454169707971248579138097910)
(12827232935949294200852, 924186581270053576906727677097174)
(33065436679783291630052, 5631909892631552419920321289160774)
(7372674555404348950052, -19747937680530464700023345469226)
(22607831963797359224228, 2960352575961560916524520374952662)
(737204539032717337285684/49, 460303105192457435627209454201627290/343)
(4010191100046861427000778/289, 5494491359021592635302644883361509162/4913)
(1946634581451279449312089508234452/272785799521, 1838459638575007135224268821574744478506823348086/142473022446023569)
(3299025083229335262330935717362324533212/448412170265950801, -5202261120756861802535398171563325259886541140503614347194/300272864337972520419865049)
(3522474669138848852416043/484, -29346332102761153412936864135683123/10648)
(35607385458471587955074641516990447337138324141362036295828/4986832524114661851201128496745323169, 136182286892595693837902890550962941476765817272979742626941282870777401113639110591818/11136203920632409705871206067416180410690525483295394447)
(3979128264344052310009582992261386928543698372/615283092317253914255641, 53398032202058873069609957941738298901522546423312001487207642067794/482627952628515211341406664874231811)
(1313619775158605896991067748240382830298081489146016/61588550267084952343750295281, -40761930430240448637747266301816692792493823085695038734422783407328708409786/15284455506478480463522778703246972812054071)
(11048066819520082469584044799020846357332/1565046634310661121, 48496120166268815156819271677753071568593860330152582046826/1957901884408200254519185919)
(63952536597260873840396848732113548/166306549249, -16164376706689446720289234533873075074719507548784118/67820974929586943)
(1549616489218505403941349571291811465927708752/212557907315889897796321, -582913369998543828946453544448133289639413323406123215436396071014/97997703516945355612869193481596369)
(4616313307338049609277498/49, -9834871629096703723650411236609756668/343)
(12072059801554007819289642802052/1958151001, 13281565513119644465180571165667711879774654274/86650139945251)

Commentary

2-isogenous image of D-z2old141516171819-010, rational kernel x=-3636882190631092911148. Source: https://dujella.github.io/tors/z2old141516171819.html. Independent witness points transported by the degree-2 isogeny. Witnesses saturated by exact rational halves of published and transported relations; quadratic-character independence is certified.

last edited by Natanael Wildner Fraga at 2026-09-26 20:44:04 UTC · history