curve #3771

y² + xy = x³ − 7651939281551405353097463306644353155846458299568780x + 168866449327285243262963418318599657822663828954929862351475478134667389163152

a-invariants: [1, 0, 0, -7651939281551405353097463306644353155846458299568780, 168866449327285243262963418318599657822663828954929862351475478134667389163152]
rank (lower bound): ≥ 29
torsion subgroup: trivial
conductor (N): 69821470093677383248637492361571707179381506969731854249333221538630452565545609505747913798458685760497702180730606590
discriminant (Δ): 16355552742399711498740475359081387641155546056364928780172057221321293438400401826419690955710645761125395774752142593424143019102961814595560975042304000000
Faltings height: 28.8694
naive height: 370.0140
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 31, 41, 47, 89, 59069, 56876524991098627226433672907973968800409930007103788751005557291212922669955906848539806618253461
regulator: 105483420860255864307355942817406.138200359166055324646395611598298416606106526045600881846651189
submitted by: Ritik Jain
submitted at: 2026-10-01 19:42:30 UTC
last updated: 2026-10-01 19:42:30 UTC

Witness: 29 independent points log in to add more points →

(10025630704606770575771498776/529, -2130224583667842139583358963222338524335124/12167)
(987946302814894169037080056/49, 51772482358867427127010343180983582225804/343)
(1307379694544626435804082278264/3481, -1456203545181899231960264829793424780892926788/205379)
(135972774063873379487144824, 1281544625798462369526329479246666726708)
(177145987273734372749541240409/256, 73979481000992296883459420247420264224026933/4096)
(339102327345566710621231747/4, -2878248492443497274273364117601947853177/8)
(-3510844914200870843926551368/49, -202717783960593361405037494850484847378772/343)
(15187359429918194519040574, -236974156765360675045265680368167123792)
(-74023908177615812368980536, 574174197987535441334820996768615206788)
(78799441309282376627089624, 234928443600991207409938625586617835508)
(127936482446139323657359192, -1133108903826489950048297613900222520460)
(258428952668358921468553144, 3930737425266870132033955041908366513428)
(-6353706177666182577731648, -466077382789422794380661509589835841100)
(-49931688499595150931792086, -653033180285764282835920148610890183552)
(-3083247908241953314688976954164/49729, 7056963312224838851544388848737614063028060656/11089567)
(-250447742822280291796114643/16, -34155073855394465659515110173839250572053/64)
(118601023090522810959362137448127652376635144/1365267552864182881, 637499447780113434931041880596309625543747351234636258391502753668/1595242286210444416514152721)
(2925281680755536211355988193879335032763733365807742390504964969493441755704/1549087872173687730816434479079988080792364944897401, 23959102798317486368400623722576269702325107661730084989167099743591422576606459636426778459134681136681036861079492/60969703186310759831489632035090311012708116627103811681123444236398534275149)
(-3474942116124832374235685648821504775186561257585350176/36538391022207626080646428921, 1332619926147924066657878406113968824426358014562328831781918464031140155950168868/6984319992914192858187312319831791373872781)
(74811423093461197318886818871172648426343404840352579891136289/670830741829017593238169903459617856, 460507476955092742290011295657672538814220100365293186684778890684454485190651666733220127363/549438939063980361891014484746428216835143093404586496)
(-3633349950982654409117617748056592317649354839554463064/50608689162098069165859490369, 6718032427909770638491391116493569732224156965219889406461838783068337181549644084/11385120535637633095692977054850576351565153)
(-29407159727025639603435501744188717515055036/416029217020686361, 160235965984265336375368374508789962704836874688409887438799671512/268340204977853143468714109)
(-3342040030233262122929565726716300620072442635068437544/44027947980506013485175015649, 5162997282386962824099286537980729157852557692207060954367534656629453546464134796/9238312886909486989855864631137218635874607)
(19045638861430966747745709388814234682785403573362753945791674812296/260593657247381186317595497086335122729369, 347145212005172138399477925806553157107120212423721434858340072513413172091168814277478623892288796/133028826926926776931892671045248346124724779976137102776545347)
(-708908003511750235243862584158145066471787807376877932929524232/7424377395451697042406797792634689689, 3442702466710773830512668929274852924374329328443718258592692021661794426349656886323215632580/20229728755614412254796536893839847604358472084185239613)
(-435876552610105367531552620422970673298740285305544/6954296063247725322405025, 11631215967323119585521837806576829968613265124421254627600106099025202448556/18339173691180381769492590105981631375)
(1487250011291936857774551152339233231517550118549536329645658919889335614553766070985707038819856922724314/69453914068820090184480129561037083080050445739506433232854490055403295174534081, 70489681525244808538736479263614305525347820032486088924064577728853717386984919014160919846848504077796141812990957353099324614434728877619147400744323562112/578822078043627930732300232780747012601494576495178204629831643021952662774420020920976233692415010056011007541237709729)
(-51920164509276087349598783934249627025981631407752824981253821062378496/780235858074461220054426081269966861146830025, 13494616100731403947906134189709429259625878976411946511686701724209573736625495487270826667127551660011652/21794095951485585543692724479063958673420628166191129311275860181125)
(-30456769287545392366966911461784620079809744814233/1565007316710697918214244, 1090794996662920666114659193551127492719072326745245138638529370002761227563/1957828104319585317150046838609685528)

Commentary

Specialization at T = 1969/15181 of the rank-16 fibration of X1008 given by: y^2 + (Lx+B)y = x(x+D)(x+E), where B = pE + qD + pq(L-p-q), L = 28139T^2 + 30542T - 534071, p = 3500T^2 + 607734T - 29070, q = 3807T^2 + 8441T - 3536, D = -12(6076000T^4 - 261217124T^3 - 1396653485T^2 + 819776499T + 4240150200), E = 7(346558177T^3 + 994005376T^2 + 82494799T - 248810640).

last edited by wgxli at 2026-10-02 04:47:05 UTC · history