curve #38

y² + xy = x³ − 26175960092705884096311701787701203903556438969515x + 51069381476131486489742177100373772089779103253890567848326775119094885041

a-invariants: [1, 0, 0, -26175960092705884096311701787701203903556438969515, 51069381476131486489742177100373772089779103253890567848326775119094885041]
c-invariants: [1256446084449882436622961685809657787370709070536721, -44123945595377604327137242899392065760392800145803979335441014758961586480505]
rank (lower bound): ≥ 18
torsion subgroup: ℤ/2ℤ
conductor (N): 25725142933318819706009609339333397294766021498979988120833604560475705134314952598
discriminant (Δ): 21165818183141855603806249764561281249470640134499286938938173449471997510627809748094469486747147781730447578406918725587571677363134437997346291712
Faltings height: 27.2769
naive height: 352.9804
primes of bad reduction: 2, 3, 7, 11, 17, 29, 31, 41, 47, 53, 79, 107, 109, 137, 149, 181, 313, 487, 2063, 3209, 3329, 6469, 10513, 15091, 18913, 23417, 32441, 47051, 131749, 643889
regulator: 90710216574493228744.822722175775048901778182334992644131077527874357834918
submitted by: David Renshaw
submitted at: 2026-05-27 22:12:16 UTC
last updated: 2026-07-01 22:33:41 UTC

(3231737970618117409726910, 477734049899522560689163862493530849)
(3273309887687276930053886, 677805198733135867174443509568209441)
(3241000828313547371069168, 526142335107247006178138422111656119)
(3255153282123996886918526, 595229162470140060497670577715082401)
(26966585148660027255767774, 137677903994541133602151567271381542145)
(4770216163266413128311686, 5894953964627833789313924429835166673)
(2549750315144935427762366, 950643850306503428886497351231138273)
(3713349702880881629511998, 2252141359101383750330981276127632993)
(3542121340287734605297214, 1671112741117758577475984222816297825)
(3195043262517282313487906, 228081804454697230002446209771949321)
(5646805757967450691937630, 9127716726638497548695728027642334081)
(14331333023247963581855678, 51180113307318927046718773937933570273)
(97191985119716082916503194/25, 354969759255330273188266145971062270037/125)
(1185147271535670890597935886/169, 32017173765694036331237894920457649821237/2197)
(98739463791232711776641098401842/38278969, 199867994868228186007608230146203388221790785839/236831981203)
(2294824481268463496752138910, 109931747845168723937902307597670194926241)
(-52959420769695630752203586/9, 31331086291021542344530429579191837787/27)
(154432285705268244684426598052/44521, 13312631160178214785941322691372196088408619/9393931)

Found by Elkies (2006). A curve of rank exactly 18, via Dujella's elliptic-curve rank-records tables.

last edited by David Renshaw at 2026-05-27 22:32:10 UTC · history