curve #3229

y² + xy = x³ − 13831746340469642469114774496882768456535567038x + 836300630233035697255303230267189180357740735108205713850912491994692

a-invariants: [1, 0, 0, -13831746340469642469114774496882768456535567038, 836300630233035697255303230267189180357740735108205713850912491994692]
rank (lower bound): ≥ 18
torsion subgroup: ℤ/2ℤ
conductor (N): 11148693462601817177654682417706894412080396884801101938270196382625118197171415850
discriminant (Δ): -132780188637799422238364195225440470987269046570762710356645670926249056742514308027449123121549116396954719361401786422740022584525984000000
Faltings height: 25.5937
naive height: 330.9224
primes of bad reduction: 2, 3, 5, 7, 23, 37, 101, 109, 149, 157, 179, 181, 223, 251, 311, 449, 479, 769, 953, 1609, 2017, 2221, 2239, 2857, 2897, 29363, 31121, 100799, 154667, 286397
regulator: 67445612731269299073218.632244867992043517673645453705391390468425642099437
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:43:39 UTC
last updated: 2026-09-26 20:43:39 UTC

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

(39340138363714237565148188/169, 222216172655990514922898244279158573978/2197)
(261502212010658461718063723/11236, 27341339580461615660296512550757286985907/1191016)
(-1236081808887270208729316476/9409, 17933431490344468164093627891369949667690/912673)
(15856053701167129200676149164/297025, 2559776633775840042727949184214465827401698/161878625)
(-85306263808422155247718255252/3636649, -234960365175440252657346434946657195193560218/6935089643)
(611816005641554280044223160568/7856809, 335011431752747985822796808506969967279451698/22022635627)
(3689565359987175517449940318/64009, 245883807335090570707435185662878722197548/16194277)
(85635320563967468031434411203576602509640521481425769883101761536/8350073258051505340352496213314754505225, -25058265223118167977333444122223942109142786296279322732064644256308151880049311762499354309444186/763019133664369865059700162734850268658990830000493226990875)
(-5116305189898872456042977978764/40208281, 5903088506746858585871455478291981940880278730/254960709821)
(-287920495128143006321754827686/5094049, -435914999850523734693356274272606833835045488/11497268593)
(143547831239570203871862383562752/1382575489, 1171658942339886005937479889874781845010792173010/51408304407487)
(1700288405897009508654174375355088872585378934400832420724198034978198209733176/16752252947425504870602812681115233676700244234459917561, -1499061129405447178101081022732916818342945988030820786611439206113141316958986624022697108387141930203996364098362530/68566160806590215624231105464908054048137841185389714975228254234631281745165063309)
(-2725272570002096354517309366056849466648367310752776619598011584243861914104772/19951296660486159818262225618609908462634268229398265329, -1185542914332452982601768574490181812192905210928051888008114907677879741074493587349585465732969233499046144208878622/89116206146666243171596938219169505283719924730764916858589109151759847662059859017)
(1035811055141422562169402955193/7441984, -813964430572035168555723414735690549343339177/20301732352)
(638654824687488276275031959707587720258204489480969788/1161991469364222609716573636209, -499879127913216277601469989340481718565118861764327590832230163579181969124135618/1252576932940391437712035565265577989371800823)
(492505381234427114702391982386992809552/483939325901641, -10861042863255483662197456469222620404288917899362717636646/10645997817505683637061)
(-15361884580982364415583189180435323268875797971353152831027567696874926417198097813765389847951312/427466367563546122761230279871743399830929724232719037979776477907529779409, -317055765674294493383283625521036515432520429617459517446064180140368167930690549573378164712896962253299362115896969577148272512600246879357134782/8837978202720121438813816534481717279790760872033408017068009587856603787965870941873945348992089103950424583577)
(-102969187512969972088038867613901647298699741404852/1042763605531600883447616961, -1185393190400886687863183034517412924409289090519008431185982188314027545330/33672766405278995509755548348694319044959)

Commentary

2-isogenous image of D-z2old141516171819-006, rational kernel x=70317727086699237217052. 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 20:43:40 UTC · history