curve #715

y² + xy = x³ − 167900449973288294000275741419626104756126578363055305017085x + 26418612714859942951968389601280556169962381332801121405732500376054279209359269274474225

a-invariants: [1, 0, 0, -167900449973288294000275741419626104756126578363055305017085, 26418612714859942951968389601280556169962381332801121405732500376054279209359269274474225]
rank (lower bound): ≥ 30
torsion subgroup: trivial
conductor (N): 4770664328323747960628930102079487297630600156046265106512859235775530956966965174134072580399287008091112355168863315050479935714086496409629570
discriminant (Δ): 1413886969380425214138680052003043124556947828354756741539916570067505976601567132158578437162707522507483395390316718464882460929177506059038284390520397494239274804391936000000
Faltings height: 32.8659
naive height: 420.7258
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 29, 43, 61, 73, 1747, 9461, 161309, 148598336053, 10513046902699709243, 21265263188745001146177648528265061703165253701894427047413918514586447389406094298594859
regulator: 2877010600248926870141470307600108932.8277655711767454418702222604705802748383925986655425559884972
submitted by: 7fff-zip
submitted at: 2026-09-11 19:52:55 UTC
last updated: 2026-09-11 19:52:55 UTC

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

(292409732382640336661472232410, 48217512058173522797488269108265608650127975)
(1442047647378513993246690230370, -1668244073985915580659341153384412957385967985)
(1276103584200791268663554603476290/5041, 4198227984140781347182834279859390166297111907545/357911)
(-469634263575082957751907003390, 41104766921708732542448144870038852594445775)
(-423412255629472052475046545630, -146973619245399253851057745435842020439431985)
(661428709595942323781701846200, -452472669851696468167889095227753377640048015)
(-6453920917607071515764405550270/49, -73763542104707229074939733976711690551075614055/343)
(785768367255644998311024524610, -616154386036592474476069792978829604332222385)
(-124881100285388590125079076670, -213163432066363139852955475684575381203755185)
(207865253226737568888005785410, 22346754538718217258349510454032224425409615)
(247178130884770048619323337700, 4375225163724097992267508322392785453655485)
(462233389744359667195374873570, 218105637284075736419398134511770992332329615)
(89368756998912899577267148260, -110124140546107206360143770366004656756436385)
(226859528128968920940725030370, -2043812884607768075486122989531161118767985)
(405882363301500747743075775090, 158543501428943243306526441446730291834805215)
(114686498726669134435639786770, -93119110211662134497643960214578713658896385)
(48422139847474139493448084770, -135654152950607430998143911514239851658542385)
(263954042628889288820128390716, -22152803954813097643692134939152345633189323)
(7207078869401880080605377967266, -19317513694472230091748377065551748991913567153)
(227134732405622968096268478690, 715669126354651743357661736314900940446095)
(1989609114372837923518577798153970/1681, -84008833290450509485056777155577785804012251546185/68921)
(1836676709817177133445560961250, -2431841010541417381042051198039583888525374065)
(27723085848832647012347799000210, 145953724254841803818588362984860547434707618175)
(4795001256459098629648543521554/25, 4456554425405108419433266713396666280452209107/125)
(-2546567060368633224340534349562232240889670/24641564056729, -25266535091963322277498520463484718574414498561151470238759112955/122321389299832287683)
(367693797586841586585962134212148790142925151049563089384530/1694568321539355177010830638929, -31426413236191839565363380336757527813782919538275649926345729975451825337234628929094855/2205914244358170398171031696412628167652911783)
(303627713026503109921539478198696480196053315115838468210/1605948917800556406551161801, -2436064624308454433529412256902403092127872423552104055776208282031993460371275631925/64357266640439451915181440282963911948549)
(2014206530141589092613168595471843343091570/1113972591601, 2790796546692556654880841833242305965242726377434034016856656935/1175741257832683849)
(8585182927240096168935004316802198093823718615695433953141332822510117604172670/9178630989790334171738240162877964412511089507641, 23060012048758372248834217810415849761322539740005575971174689735279185387425495726677288270528897831936329145244999915/27807814985802442338068359184655835023278375041173019289463370174408126061)
(2552711034727701442029668097609827034397199300736662754921054719747108112221990070/11275092173207018617165783827369864427418084164543849, 3877419236373601742514676206180470958676021403798801389377113137742694601317500369439564539227800499829164027085997402285/1197237058769910484761439791822706086550663629562043555278585423469506093538507)

Commentary

From the K3 family: y^2 - (Lx+B)y = x(x+D)(x+E), where B = pq(L+p+q) - pE - qD, L = 446667T^2 + 471466T + 239031, p = 318552T^2 + 368554T - 72570, q = 733413T^2 - 45082T - 14960, D = 5(7174492962T^4 - 7114589515T^3 - 22069002960T^2 + 3909144679T - 205134150), E = 882769396002T^4 + 811447034567T^3 - 1174040743T^2 - 32493137198T - 2386325360; apply the two-neighbor construction with v=[0,0,−1,2,0,−1,−1,1,0,−1,0,0,0,1,0,0,0] in the class-1 section basis; the new fiber class is (3,2,−v). Set u=(−109758864t+3649521776)/(373496649t+267426299), then specialize at t=32359/40226, equivalently u=1685340892720/268746686689.

last edited by 7fff-zip at 2026-09-21 16:16:38 UTC · history