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The Rank

Conjecture 2.2   There exist elliptic curves over  $ \mathbb{Q}$ of arbitrarily large rank.

As far as I know, nobody has any real clue as to how to prove Conjecture 2.2 (Doug Ulmer recently wrote a paper which gives theoretical evidence). The current ``world record'' is a curve of rank $ \geq 24$. It was discovered in January 2000 by Roland Martin and William McMillen of the National Security Agency. For security reasons, I won't tell you anything about how they found it.

Theorem 2.3   The elliptic curve

$\displaystyle y^2 + xy + y = x^3 - 120039822036992245303534619191166796374x + 504224992484910670010801799168082726759443756222911415116 $

over $ \mathbb{Q}$ has rank at least $ 24$. The following points $ P_1,...,P_{24}$ are independent points on the curve:

$\displaystyle P_1$ $\displaystyle = (2005024558054813068, -16480371588343085108234888252)$    
$\displaystyle P_2$ $\displaystyle = (-4690836759490453344, -31049883525785801514744524804)$    
$\displaystyle P_3$ $\displaystyle = (4700156326649806635, -6622116250158424945781859743)$    
$\displaystyle P_4$ $\displaystyle = (6785546256295273860, -1456180928830978521107520473)$    
$\displaystyle P_5$ $\displaystyle = (6823803569166584943, -1685950735477175947351774817)$    
$\displaystyle P_6$ $\displaystyle = (7788809602110240789, -6462981622972389783453855713)$    
$\displaystyle P_7$ $\displaystyle = (27385442304350994620556, 4531892554281655472841805111276996)$    
$\displaystyle P_8$ $\displaystyle = (54284682060285253719/4, -296608788157989016192182090427/8)$    
$\displaystyle P_9$ $\displaystyle = (-94200235260395075139/25, -3756324603619419619213452459781/125)$    
$\displaystyle P_{10}$ $\displaystyle = (-3463661055331841724647/576, -439033541391867690041114047287793/13824)$    
$\displaystyle P_{11}$ $\displaystyle = (-6684065934033506970637/676, -473072253066190669804172657192457/17576)$    
$\displaystyle P_{12}$ $\displaystyle = (-956077386192640344198/2209, -2448326762443096987265907469107661/103823)$    
$\displaystyle P_{13}$ $\displaystyle = (-27067471797013364392578/2809, -4120976168445115434193886851218259/148877)$    
$\displaystyle P_{14}$ $\displaystyle = (-25538866857137199063309/3721, -7194962289937471269967128729589169/226981)$    
$\displaystyle P_{15}$ $\displaystyle = (-1026325011760259051894331/108241, -1000895294067489857736110963003267773/35611289)$    
$\displaystyle P_{16}$ $\displaystyle = (9351361230729481250627334/1366561, -2869749605748635777475372339306204832/1597509809)$    
$\displaystyle P_{17}$ $\displaystyle = (10100878635879432897339615/1423249, -5304965776276966451066900941489387801/1697936057)$    
$\displaystyle P_{18}$ $\displaystyle = (11499655868211022625340735/17522596, -1513435763341541188265230241426826478043/73349586856)$    
$\displaystyle P_{19}$ $\displaystyle = (110352253665081002517811734/21353641, -461706833308406671405570254542647784288/98675175061)$    
$\displaystyle P_{20}$ $\displaystyle = (414280096426033094143668538257/285204544, 266642138924791310663963499787603019833872421/4816534339072)$    
$\displaystyle P_{21}$ $\displaystyle = (36101712290699828042930087436/4098432361, -2995258855766764520463389153587111670142292/262377541318859)$    
$\displaystyle P_{22}$ $\displaystyle = (45442463408503524215460183165/5424617104, -3716041581470144108721590695554670156388869/399533898943808)$    
$\displaystyle P_{23}$ $\displaystyle = (983886013344700707678587482584/141566320009, -126615818387717930449161625960397605741940953/53264752602346277)$    
$\displaystyle P_{24}$ $\displaystyle = (1124614335716851053281176544216033/152487126016, -37714203831317877163580088877209977295481388540127/59545612760743936)$    

Proof. See 
    http://listserv.nodak.edu/scripts/wa.exe?A2=ind0005&L=nmbrthry&P=R182
$ \qedsymbol$

next up previous
Next: How to Compute Up: Exploring the Possibilities Previous: The Torsion Subgroup
William A Stein 2001-11-16