curve #4070

y² + xy + y = x³ − 441395428975166888073810070214023704415261193718328773x + 324120732555947339871512064945375563392437578948692835106747039439133358282465532

a-invariants: [1, 0, 1, -441395428975166888073810070214023704415261193718328773, 324120732555947339871512064945375563392437578948692835106747039439133358282465532]
c-invariants: [21186980590808010627542883370273137811932537298479781081, -280040312928338501648986424144584957657278084152984934587639148793310027503769891485]
rank (lower bound): ≥ 20
torsion subgroup: ℤ/2ℤ
conductor (N): 6131429525853961023265331906599034669054966945113256585858859273406727832034414573630186014
discriminant (Δ): -39879625241025087593164194375504055816857517313087647708794127608108520772982930296251239372408049327177673981806841816425762139399640332231291042943185175321540828
Faltings height: 30.0772
naive height: 384.2887
primes of bad reduction: 2, 3, 7, 11, 13, 17, 19, 31, 47, 61, 71, 73, 97, 101, 151, 157, 163, 211, 317, 331, 337, 409, 431, 521, 541, 739, 761, 1637, 3433, 6397, 22037, 81157, 472859, 2737040129
regulator: 7296121240460681268714.42200590872554833194799249069269740714802441728453892453
submitted by: wgxli
submitted at: 2026-10-09 03:51:12 UTC
last updated: 2026-10-09 03:51:12 UTC

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

(-896079305954870452843526387, 363159784079992967568442975956215233370)
(-43042296185941374974509729, 18521329960270726980163298379689831813280)
(-859021921477439840023600531, 8330712785955707958638570598619348409592)
(-895084829929099000762398259, 1444243063896710577699849394520880469080)
(372088261491726618111060356, 14539538279719184793713591741255375486280)
(407562830856814350456275105, 14557595579844105186826374597881276877546)
(507192289203423638831300489, 15189488371197514590756070683824142833532)
(547821042805122347630219573, 15707337404075829720375904095998803903146)
(568011345362064169265841983, 16020753233999194972523317246074039651240)
(697504250391677475084683186, 18857090382173184954807834834874526061858)
(761772961032593099800988489, 20734825512184336678064182408318128828282)
(804480365436112376541599911, 22128654386237525219556254400035578288580)
(855296689130705625912900668, 23922247926985594246863085953409172793075)
(-194557269099799315344472126, 20065715455933393230754053784577835250923)
(-474628657528661224209280819, 20656692219121572724377928033741514984280)
(-518343353274041230288511590, 20338306574423649200623847257747390904331)
(-567289971995044886548302004, 19797875103834318364019451347863582692723)
(-775905464865785496290621732, 14123881960256940286524165863729095530800)
(1361844712068944471474046509, 47420612926951632727479946250310265232568)
(5559548004942856616393295613, 411956141695244294337953715631935347354120)

Commentary

specialization at T = -682893/97948 of the rank-9 fibration of X660 given by: y^2 + (4634T^2 + 5264T - 4032)xy = x(x + (43T - 5553)(93T - 193)(235T + 279))(x + (1 - 39T)(2220T - 961)(63T + 47)(T - 9)).

last edited by wgxli at 2026-10-09 03:51:12 UTC · history