curve #630

y² = x³ + x² − 478084759064998390933143695734260558414948830620x + 119142731212438304178867546335237459944511190723779503318854859356288944

a-invariants

[0, 1, 0, -478084759064998390933143695734260558414948830620, 119142731212438304178867546335237459944511190723779503318854859356288944]

rank (lower bound)

≥ 27

torsion subgroup

trivial

conductor (N)

625847273392133749141084531784604199332000011136824937628804232804540038348472590036929315025313329802578398367238396088

discriminant (Δ)

861265623558584942823941824946868543987257828061527080055139637101120161140597943176655121328773583392980179877870893149948554653730382857870592

Faltings height

26.3656

naive height

340.9720

primes of bad reduction

2, 3, 7, 11, 13, 23, 37, 1223, 24437972051, 561063059024179810500844410439, 1825535106347063733958801675489652807861250517266415081493807015950621

regulator

24236510207521892374816692833903.861518567390024375340731991595394554991956747072736883155144

submitted by

Roy van Rijn

submitted at

2026-09-07 08:28:06 UTC

later contributions

primes of bad reduction recorded · Roy van Rijn · 2026-09-07 08:42:04 UTC

Witness: 27 independent points log in to add more points →

(53667352945221110797066706, -393123840937546600904924270801302945074)
(17433140163317800903227686/25, -44166876215574006987475020480154732266/125)
(-1721248467787499383412042/9, -12182390159075324580780103175160717302/27)
(1157968384363419683673312958/5041, -52410317928157597729226268883064470355430/357911)
(13314547971206989228789570/81, -154640768961912986139763843299284745202/729)
(-7986443927573349154539106/25, 61143901852462354572971542193470372878/125)
(-3642222329410685466032117722/9409, -452858549349607077540383669747675126667618/912673)
(-721845154526690289151625, 296851736509934232919179997909579962)
(2894867990351394076055090, 4795299663563472926167554165441593838)
(433102838826316746957623402/1681, -7879041312422614432949217932806790935386/68921)
(-128074317074627888032189, 422223049204293004059837703440360024)
(1501660173050891496579349193/3136, -554483402212345171121593714008446040795/175616)
(4424868099066997549873823771194/26245129, 27988089409984321151212127980896182989794564942/134453795867)
(104895067831791542144210, 264854971289458866152756551081690062)
(64595182975169807860676360/121, -168675696774150469034295046391223306228/1331)
(575097304483033122291842, -185481482033510125180718655306986334)
(45674269578896212182153533234/93025, -1493606770913985246141323930633937839417202/28372625)
(-7908835375190412731535529645/10609, 270431092357907806919042797577763321642624/1092727)
(-40258437569358328664876032138/208849, 43123082670589542141031190857985024555047710/95443993)
(227213990907893991470810, 149148850212614015881816024530642438)
(644516148247215773856407825/1024, 8518215494606766190616647848620965751559/32768)
(83470981518157502874820424/529, -2655481912630607309209690563065943819108/12167)
(-105141436805975748366562, 410179156429245433039211878735575030)
(-179369262832116806775606830/289, 2065706816417688350831704375501461650094/4913)
(482528797938179839881914, -28323081641884004367139543755211098)
(6455253296606632615676474465090/6507601, -13081723379415275642796963889713878081569537038/16600890151)
(188984785648092784299290, 188524863650845064420491293984766662)

Commentary

This curve was found prospectively as the specialization t = 2818/1535 of an elliptic K3 family with generic Mordell–Weil rank 17. We found 27 independent rational points on the specialized curve, proving rank at least 27 over Q; the exact rank is not currently known. The submitted equation is a globally minimal integral Weierstrass model. In this presentation, the specialization has 10 independent directions beyond the 17 generic sections.

last edited by Roy van Rijn at 2026-09-07 08:28:06 UTC · history