curve #684

y² + xy = x³ − 3548762533798994120799398878730631531519537372413248580634134663x + 91542842811256540806273969630813313608082057896687957787072591727829337872800980019462021870217

a-invariants: [1, 0, 0, -3548762533798994120799398878730631531519537372413248580634134663, 91542842811256540806273969630813313608082057896687957787072591727829337872800980019462021870217]
rank (lower bound): ≥ 30
torsion subgroup: trivial
conductor (N): 11903707897514041200039733573485179202022772635106127382433543310087968603272764429442025598562332589761026235376461727399362018417134689123922641580850
discriminant (Δ): -759905002814538994665751102688897294043900890039775285760878271432682613678540949688476169694259899877190567410305232170882135569308737160050608292308387718686152666395074572858481735761920000
Faltings height: 35.5475
naive height: 450.8376
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 23, 61, 167, 227, 197807, 38920557961030715245659713049969557675284553, 11112574756186583618179400877725039607532757088496538476826670632729946132591380269
regulator: 50228098817791483446761312102573629507.148162950943905263851409240873691049257065190772120529924165
submitted by: wgxli
submitted at: 2026-09-09 01:41:29 UTC
last updated: 2026-09-09 01:41:29 UTC

Witness: 30 independent points log in to add more points →

(-826992051754534703281045407617427122/11881, 46949793374125100720556992994755237587402576477015513/1295029)
(1110904784358357426257172083369693630552265612022/72726671889526561, 3966420836654434890537719982789602471209951532202377991340764516842255339/19612843857688084113930191)
(9899409582183466950210116522935893251722225222/59751338550201, -929639494484516807621592832062573889540413737820809027144581094693569/461871812107860159699)
(-776109391919217739972187817828401498/11881, 273536173624581409936354771150828350617244392256321961/1295029)
(-2631053160533267635857461603029331250174845708762/85665857337413809, -10385458654257186641598991687114964748166890862723905111825686985806735117/25073308743270408751167127)
(-8536916401937548905438319542468444/1681, 22799642452473863712306362556862689719964239630615199/68921)
(292266820037482737938549034810358, -4900995625076800240142446875409699189970755225883)
(-37299992141745304310527287026845253/5476, 137656534005147056601811659516873660230984465081534831/405224)
(286872012674832091812704929270318/9, 2807441940722430226033552899803940562680223395583/27)
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(-36411655143900841943432710005796658/10201, 332524584388453613823716634120636359179907735844870809/1030301)
(107855422775515228430208312325558, 981554523483629057034989806895888046588689201317)
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(1348614902481510154279972518387536507249589543641565617281342/49481139795276341646536388921, 1351059949256968151210806553469169529286207748835183192004102204236981258604968069764077649/11006761156941500784253160040601241709212781)
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(-2499529035548534736445589811069126072294272256298541050514/207283433219497904332864441, -1086649649648086086413066041526669460548127312489921445855677387828076652988613912256561/2984330310219089850084972049671941069011)
(49793768318602640967360637071646262182/9406489, -7789719035200054340444841306964987307817462410884065233105/28849701763)
(411445124993187451613497585457294238951844198336374694/161868504637437519769, 263845559528507726144895578536513788167084603188250487324484031180252521713844207/2059413389361592686516963745603)
(8117473604583108584217284948888134922623993542/226946059843441, 348165212033335251813427523232140713683132673167465257402346065596221/3418880889159221092489)
(3040453952458257711445728047319478447998/169963369, -407276440904300170926480921852310637688678264338385343565703/2215812441653)
(2804589318436316129949874702142311018169250502/258261580929361, -967013543577517380789570736163424278117020853904516306119978016139789/4150397643295333608359)
(6876694020757382371422341018170701156311438693005331158/4649310479378924085169, 18018688195026484301197298240412055666835044982306672759930194375588090100005375187/317017198513194573048104858465303)
(703424377452628123024977066077900866448901142/7026364231441, -16017886294728818433623428662054436805193871160948680709432002990979/18624987432843370489)
(52406850748249798332013102203464286382300897/1208997007936, -184021006072446648957712759616927706604823391508081650182628351379/1329345406093981184)
(-2737226004963971283248797413615961658/167281, 26073443699418636883566223518658660131890517397807706941/68417929)
(43123537046181193582378623666605915587139358149053153786/1419984702465211095423721, 183645770582867570010188898397857428280461931291074058052976712179196762601064787505/1692097947268258300874462452381156981)
(23960398545398502335779303304619855634040276946118/506588465350580089, -61931582780192453031341037295667026194083124530881822489899402105338863457/360564483577811072559915187)
(81116147797677665958820866482866041629386/1957974001, -10830735992335787385793092980055102915238243905085271833250059/86638391570249)
(83702376039096122412188533563852396822386084450084062/442971361931253292201, -23153768378715783968413132768791083676313948333321332951779334076990308084606361/9323167246857733019069103347051)
(25569855349101742873057450314998097562/249001, 111826642151744631022239861549805397137875693121716743011/124251499)

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