curve #37

y² + xy + y = x³ − 957089489055751752507625259831765957846101x + 351598252970651757672333752869879740192822872602430248013582348

a-invariants: [1, 0, 1, -957089489055751752507625259831765957846101, 351598252970651757672333752869879740192822872602430248013582348]
c-invariants: [45940295474676084120366012471924765976612825, -303780890566643118628965272922788109652779510947207621432700068125]
rank (lower bound): ≥ 17
torsion subgroup: ℤ/2ℤ
conductor (N): 14736232836353426730586889330999407111621750064300242106070930802493850
discriminant (Δ): 2705241843717540804757209194175752043961253605237251059078988771418272961322642530579319128322487000408815513923180322592562500
Faltings height: 23.0393
naive height: 301.6077
primes of bad reduction: 2, 3, 5, 7, 17, 23, 37, 41, 67, 79, 97, 101, 103, 151, 229, 389, 857, 977, 983, 1499, 1531, 11719, 12907, 49639, 938573, 4153367
regulator: 471391564634837735.979728668453443736342367256126191005605987032331435932
submitted by: David Renshaw
submitted at: 2026-05-27 22:12:15 UTC
last updated: 2026-07-01 22:36:41 UTC

(1037048102780198794447, 21779881979625846052436063081576)
(640151922319155456727, 1116379497785052017163204160436)
(8215515531545032362283/4, 671712937153262899818205610666563/8)
(-252494436924143857397/4, 162335292364600094400633862613143/8)
(3033356276097950346763/4, 62943339550041852920918618404183/8)
(-697039650804322492943, 26077977524885678965262644558346)
(11414518769372982291382, 1215170223607965741833261815562846)
(891674633090402503696, 14392420123651702644251835214211)
(94919125996596744048847, 29242036615099659730101568678875176)
(636565095929048595313, 542487929999796513742831489802)
(4202001410067480893647, 265563310660870839720550898955176)
(438987621400369262071, 4005643151170292277973367502092)
(-1055345243070301537073, 13647840402839172835783080260936)
(-1120995419674665243803, 3976668559063776088092945926576)
(476001270991734096802, 1968115763191782636541113312386)
(639949947842762264623, 1091505880400660592954070253744)
(45374293429909818890233/64, 2776529523851458968917039502401287/512)

Found by Elkies (2005). A curve of rank exactly 17, via Dujella's elliptic-curve rank-records tables.

last edited by David Renshaw at 2026-05-27 22:32:09 UTC · history