History of Covering radii and covering densities of the classical lattices

 Display properties:   number-header: $R/\rho$ or $\Theta(L)$-Numbers: []+Numbers:+- params:+    family: Z+    n: '1'+    expression: radius+  number: '1'+  comment: $R/\rho=1$; $R^2=\frac{1}{4}$ with $\mu=1$+  equals: HREF{One}+- params:+    family: Z+    n: '1'+    expression: density+  number: '1'+  comment: $\Theta(\mathbb{Z})=1$; the thinnest covering of $\mathbb{R}$ by equal+    intervals+  equals: HREF{One}+- params:+    family: Z+    n: '2'+    expression: radius+  number: '1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573'+  comment: $R/\rho=\sqrt{2}$; $R^2=\frac{1}{2}$ with $\mu=1$+  equals: HREF{Algebraic_numbers_of_degree_2#1,0,-2,2}+- params:+    family: Z+    n: '2'+    expression: density+  number: '1.570796326794896619231321691639751442098584699687552910487472296153908203143104499314017412671058534'+  comment: $\Theta(\mathbb{Z}^{2})=\frac{\pi}{2}$+  equals: HREF{Rational_multiples_of_pi#1/2}+- params:+    family: Z+    n: '3'+    expression: radius+  number: '1.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576'+  comment: $R/\rho=\sqrt{3}$; $R^2=\frac{3}{4}$ with $\mu=1$+  equals: HREF{Algebraic_numbers_of_degree_2#1,0,-3,2}+- params:+    family: Z+    n: '3'+    expression: density+  number: '2.720699046351326775891117386463233598426099372139110863354827403082184771689530825526187482318090253'+  comment: $\Theta(\mathbb{Z}^{3})=\frac{\sqrt{3}\,\pi}{2}$+- params:+    family: Z+    n: '4'+    expression: radius+  number: '2'+  comment: $R/\rho=2$; $R^2=1$ with $\mu=1$+- params:+    family: Z+    n: '4'+    expression: density+  number: '4.934802200544679309417245499938075567656849703620395313206674688110022411209602621500886701859276116'+  comment: $\Theta(\mathbb{Z}^{4})=\frac{\pi^{2}}{2}$+- params:+    family: Z+    n: '5'+    expression: radius+  number: '2.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275'+  comment: $R/\rho=\sqrt{5}$; $R^2=\frac{5}{4}$ with $\mu=1$+  equals: HREF{Algebraic_numbers_of_degree_2#1,0,-5,2}+- params:+    family: Z+    n: '5'+    expression: density+  number: '9.195460979944543879309937184408385776826125935585245624246320340698927141124778745877273000137677141'+  comment: $\Theta(\mathbb{Z}^{5})=\frac{5\sqrt{5}\,\pi^{2}}{12}$+- params:+    family: Z+    n: '6'+    expression: radius+  number: '2.449489742783178098197284074705891391965947480656670128432692567250960377457315026539859433104640235'+  comment: $R/\rho=\sqrt{6}$; $R^2=\frac{3}{2}$ with $\mu=1$+- params:+    family: Z+    n: '6'+    expression: density+  number: '17.44103063266864884870542722524453480125172481831037307795630268339109714107445964613126933083912348'+  comment: $\Theta(\mathbb{Z}^{6})=\frac{9\pi^{3}}{16}$+- params:+    family: Z+    n: '7'+    expression: radius+  number: '2.645751311064590590501615753639260425710259183082450180368334459201068823230283627760392886474543611'+  comment: $R/\rho=\sqrt{7}$; $R^2=\frac{7}{4}$ with $\mu=1$+- params:+    family: Z+    n: '7'+    expression: density+  number: '33.49758301440499875246683298329190753168095765962112073162977089625749335692405662733881463813925684'+  comment: $\Theta(\mathbb{Z}^{7})=\frac{49\sqrt{7}\,\pi^{3}}{120}$+- params:+    family: Z+    n: '8'+    expression: radius+  number: '2.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145'+  comment: $R/\rho=2\sqrt{2}$; $R^2=2$ with $\mu=1$+- params:+    family: Z+    n: '8'+    expression: density+  number: '64.93939402266829149096022179247007416648505711512361446097857292664723697121813079341457815650199503'+  comment: $\Theta(\mathbb{Z}^{8})=\frac{2\pi^{4}}{3}$+- params:+    family: Z+    n: '9'+    expression: radius+  number: '3'+  comment: $R/\rho=3$; $R^2=\frac{9}{4}$ with $\mu=1$+- params:+    family: Z+    n: '9'+    expression: density+  number: '126.8057631496210299024375045179750466090203749203351293090715526701584886214768321832122164538123778'+  comment: $\Theta(\mathbb{Z}^{9})=\frac{729\pi^{4}}{560}$+- params:+    family: Z+    n: '10'+    expression: radius+  number: '3.162277660168379331998893544432718533719555139325216826857504852792594438639238221344248108379300295'+  comment: $R/\rho=\sqrt{10}$; $R^2=\frac{5}{2}$ with $\mu=1$+- params:+    family: Z+    n: '10'+    expression: density+  number: '249.0394570192720160015798421577438203778488823470662358019958037811157162545687048224149580676311189'+  comment: $\Theta(\mathbb{Z}^{10})=\frac{625\pi^{5}}{768}$+- params:+    family: Z+    n: '11'+    expression: radius+  number: '3.316624790355399849114932736670686683927088545589353597058682146116484642609043846708843399128290651'+  comment: $R/\rho=\sqrt{11}$; $R^2=\frac{11}{4}$ with $\mu=1$+- params:+    family: Z+    n: '11'+    expression: density+  number: '491.3994429783274077305032733510548573679085060649560551376626397784182814447335498269596449321539448'+  comment: $\Theta(\mathbb{Z}^{11})=\frac{14641\sqrt{11}\,\pi^{5}}{30240}$+- params:+    family: Z+    n: '12'+    expression: radius+  number: '3.464101615137754587054892683011744733885610507620761256111613958903866033817600074162292373514497151'+  comment: $R/\rho=2\sqrt{3}$; $R^2=3$ with $\mu=1$+- params:+    family: Z+    n: '12'+    expression: density+  number: '973.4065584949957424930971866980751476030577471820596986403396398276450867528795911679616767663146829'+  comment: $\Theta(\mathbb{Z}^{12})=\frac{81\pi^{6}}{80}$+- params:+    family: Z+    n: '13'+    expression: radius+  number: '3.605551275463989293119221267470495946251296573845246212710453056227166948293010445204619082018490718'+  comment: $R/\rho=\sqrt{13}$; $R^2=\frac{13}{4}$ with $\mu=1$+- params:+    family: Z+    n: '13'+    expression: density+  number: '1934.564465080156808372399720669519650526715713110823667474564565810867384341276384000127792539696079'+  comment: $\Theta(\mathbb{Z}^{13})=\frac{371293\sqrt{13}\,\pi^{6}}{665280}$+- params:+    family: Z+    n: '14'+    expression: radius+  number: '3.741657386773941385583748732316549301756019807778726946303745467320035156306939027976809895194379572'+  comment: $R/\rho=\sqrt{14}$; $R^2=\frac{7}{2}$ with $\mu=1$+- params:+    family: Z+    n: '14'+    expression: density+  number: '3855.625845862758354502809024559816075122113859440416458450227146206417214327540626958854303970448735'+  comment: $\Theta(\mathbb{Z}^{14})=\frac{117649\pi^{7}}{92160}$+- params:+    family: Z+    n: '15'+    expression: radius+  number: '3.872983346207416885179265399782399610832921705291590826587573766113483091936979033519287376858673518'+  comment: $R/\rho=\sqrt{15}$; $R^2=\frac{15}{4}$ with $\mu=1$+- params:+    family: Z+    n: '15'+    expression: density+  number: '7703.081221506743320238474181423334429672389139837316704348374248648010622365430603166596024704184369'+  comment: $\Theta(\mathbb{Z}^{15})=\frac{84375\sqrt{15}\,\pi^{7}}{128128}$+- params:+    family: Z+    n: '16'+    expression: radius+  number: '4'+  comment: $R/\rho=4$; $R^2=4$ with $\mu=1$+- params:+    family: Z+    n: '16'+    expression: density+  number: '15422.62819120042505285660526349290186637763774438169631225513271274367415154814178911161185851963348'+  comment: $\Theta(\mathbb{Z}^{16})=\frac{512\pi^{8}}{315}$+- params:+    family: Z+    n: '17'+    expression: radius+  number: '4.123105625617660549821409855974077025147199225373620434398633573094954346337621593587863650810684297'+  comment: $R/\rho=\sqrt{17}$; $R^2=\frac{17}{4}$ with $\mu=1$+- params:+    family: Z+    n: '17'+    expression: density+  number: '30936.18965664557911254280529800617630643392676764917432691134470204987726982785398091695548836627243'+  comment: $\Theta(\mathbb{Z}^{17})=\frac{410338673\sqrt{17}\,\pi^{8}}{518918400}$+- params:+    family: Z+    n: '18'+    expression: radius+  number: '4.242640687119285146405066172629094235709015626130844219530039213972197435386321116551162602982924718'+  comment: $R/\rho=3\sqrt{2}$; $R^2=\frac{9}{2}$ with $\mu=1$+- params:+    family: Z+    n: '18'+    expression: density+  number: '62158.20226605830320885917004989834733191992206012469738132381874101551170378363722030136547961342968'+  comment: $\Theta(\mathbb{Z}^{18})=\frac{4782969\pi^{9}}{2293760}$+- params:+    family: Z+    n: '19'+    expression: radius+  number: '4.358898943540673552236981983859615659137003925232444936890344138159557328203158085656159155851944527'+  comment: $R/\rho=\sqrt{19}$; $R^2=\frac{19}{4}$ with $\mu=1$+- params:+    family: Z+    n: '19'+    expression: density+  number: '125076.7174480546452783241250492294740758993177905588649085842250941399772275662045993518655132078002'+  comment: $\Theta(\mathbb{Z}^{19})=\frac{16983563041\sqrt{19}\,\pi^{9}}{17643225600}$+- params:+    family: Z+    n: '20'+    expression: radius+  number: '4.472135954999579392818347337462552470881236719223051448541794490821041851275609798828828816757564550'+  comment: $R/\rho=2\sqrt{5}$; $R^2=5$ with $\mu=1$+- params:+    family: Z+    n: '20'+    expression: density+  number: '252020.4237306060548105302173134653286834613383608074803886950441207704036713648032768431470042627452'+  comment: $\Theta(\mathbb{Z}^{20})=\frac{390625\pi^{10}}{145152}$+- params:+    family: Z+    n: '21'+    expression: radius+  number: '4.582575694955840006588047193728008488984456576767971902607242123906868425547770886604361559493445033'+  comment: $R/\rho=\sqrt{21}$; $R^2=\frac{21}{4}$ with $\mu=1$+- params:+    family: Z+    n: '21'+    expression: density+  number: '508417.4439562997111121903708341768348796453237360756387586965398546909507526642147414598426410216698'+  comment: $\Theta(\mathbb{Z}^{21})=\frac{1400846643\sqrt{21}\,\pi^{10}}{1182438400}$+- params:+    family: Z+    n: '22'+    expression: radius+  number: '4.690415759823429554565630113544466280588228353411737153605701891017024632753239721482115596061543135'+  comment: $R/\rho=\sqrt{22}$; $R^2=\frac{11}{2}$ with $\mu=1$+- params:+    family: Z+    n: '22'+    expression: density+  number: '1026791.975705456089383328761377033940244368664939119091653073681301466843464298285412317544519609581'+  comment: $\Theta(\mathbb{Z}^{22})=\frac{25937424601\pi^{11}}{7431782400}$+- params:+    family: Z+    n: '23'+    expression: radius+  number: '4.795831523312719541597438064162693919996707041904129346485309114448257235907464082492191446436918861'+  comment: $R/\rho=\sqrt{23}$; $R^2=\frac{23}{4}$ with $\mu=1$+- params:+    family: Z+    n: '23'+    expression: density+  number: '2075773.691736594681880477768612226317390481007737044003588831670484793890888867034679469930329289673'+  comment: $\Theta(\mathbb{Z}^{23})=\frac{41426511213649\sqrt{23}\,\pi^{11}}{28158588057600}$+- params:+    family: Z+    n: '24'+    expression: radius+  number: '4.898979485566356196394568149411782783931894961313340256865385134501920754914630053079718866209280470'+  comment: $R/\rho=2\sqrt{6}$; $R^2=6$ with $\mu=1$+- params:+    family: Z+    n: '24'+    expression: density+  number: '4200263.272709858379779284458335555642362973511006520171785967060591687153098661895533230276901857311'+  comment: $\Theta(\mathbb{Z}^{24})=\frac{8748\pi^{12}}{1925}$+- params:+    family: A+    n: '1'+    expression: radius+  number: '1'+  comment: $R/\rho=1$; $R^2=\frac{1}{2}$ with $\mu=2$; $A_1=\sqrt{2}\,\mathbb{Z}$+  equals: HREF{One}+- params:+    family: A+    n: '1'+    expression: density+  number: '1'+  comment: $\Theta(A_{1})=1$; the thinnest covering of $\mathbb{R}$ by equal intervals;+    $A_1=\sqrt{2}\,\mathbb{Z}$+  equals: HREF{One}+- params:+    family: A+    n: '2'+    expression: radius+  number: '1.154700538379251529018297561003914911295203502540253752037204652967955344605866691387430791171499050'+  comment: $R/\rho=\frac{2\sqrt{3}}{3}$; $R^2=\frac{2}{3}$ with $\mu=2$; $A_2$ is+    the hexagonal lattice, $A_2^{*}\cong A_2$+  equals: HREF{Algebraic_numbers_of_degree_2#3,0,-4,2}+- params:+    family: A+    n: '2'+    expression: density+  number: '1.209199576156145233729385505094770488189377498728493717046589956925415454084235922456083325474706779'+  comment: $\Theta(A_{2})=\frac{2\sqrt{3}\,\pi}{9}$; the thinnest covering of the+    plane by equal discs CITE{Kershner}; $A_2$ is the hexagonal lattice, $A_2^{*}\cong+    A_2$+- params:+    family: A+    n: '3'+    expression: radius+  number: '1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573'+  comment: $R/\rho=\sqrt{2}$; $R^2=1$ with $\mu=2$; $A_3=D_3$, the face-centred cubic+    lattice+  equals: HREF{Algebraic_numbers_of_degree_2#1,0,-2,2}+- params:+    family: A+    n: '3'+    expression: density+  number: '2.094395102393195492308428922186335256131446266250070547316629728205210937524139332418689883561411379'+  comment: $\Theta(A_{3})=\frac{2\pi}{3}$; $A_3=D_3$, the face-centred cubic lattice+  equals: HREF{Rational_multiples_of_pi#2/3}+- params:+    family: A+    n: '4'+    expression: radius+  number: '1.549193338482966754071706159912959844333168682116636330635029506445393236774791613407714950743469407'+  comment: $R/\rho=\frac{2\sqrt{15}}{5}$; $R^2=\frac{6}{5}$ with $\mu=2$+- params:+    family: A+    n: '4'+    expression: density+  number: '3.177951314668834364689514290931538124471109123338260887739528309745549219972723534575185548847581220'+  comment: $\Theta(A_{4})=\frac{18\sqrt{5}\,\pi^{2}}{125}$+- params:+    family: A+    n: '5'+    expression: radius+  number: '1.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576'+  comment: $R/\rho=\sqrt{3}$; $R^2=\frac{3}{2}$ with $\mu=2$+  equals: HREF{Algebraic_numbers_of_degree_2#1,0,-3,2}+- params:+    family: A+    n: '5'+    expression: density+  number: '5.921762640653615171300694599925690681188219644344474375848009625732026893451523145801064042231131339'+  comment: $\Theta(A_{5})=\frac{3\pi^{2}}{5}$+- params:+    family: A+    n: '6'+    expression: radius+  number: '1.851640199545102923133133553167999045058629802016670442774672685088462107955441248971531687601602924'+  comment: $R/\rho=\frac{2\sqrt{42}}{7}$; $R^2=\frac{12}{7}$ with $\mu=2$+- params:+    family: A+    n: '6'+    expression: density+  number: '9.840087624865802147789218335069302113021733263491452817152078489189529621291259569064160629449402174'+  comment: $\Theta(A_{6})=\frac{288\sqrt{7}\,\pi^{3}}{2401}$+- params:+    family: A+    n: '7'+    expression: radius+  number: '2'+  comment: $R/\rho=2$; $R^2=2$ with $\mu=2$+- params:+    family: A+    n: '7'+    expression: density+  number: '18.89906388132560467838556346947132659945160445920616088024048036803437404493147796575494158177699730'+  comment: $\Theta(A_{7})=\frac{64\pi^{3}}{105}$+- params:+    family: A+    n: '8'+    expression: radius+  number: '2.108185106778919554665929029621812355813036759550144551238336568528396292426158814229498738919533530'+  comment: $R/\rho=\frac{2\sqrt{10}}{3}$; $R^2=\frac{20}{9}$ with $\mu=2$+- params:+    family: A+    n: '8'+    expression: density+  number: '32.99263020000421251382422486027032168190065392222913908498631962945040744358996636356987154219478486'+  comment: $\Theta(A_{8})=\frac{20000\pi^{4}}{59049}$+- params:+    family: A+    n: '9'+    expression: radius+  number: '2.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275'+  comment: $R/\rho=\sqrt{5}$; $R^2=\frac{5}{2}$ with $\mu=2$+  equals: HREF{Algebraic_numbers_of_degree_2#1,0,-5,2}+- params:+    family: A+    n: '9'+    expression: density+  number: '64.42400200661536854261926765125999421278279475706707783827239377643575096351005039029224023462499507'+  comment: $\Theta(A_{9})=\frac{125\pi^{4}}{189}$+- params:+    family: A+    n: '10'+    expression: radius+  number: '2.335496832484568912748526995934534621721336820283684340052085678956094491777786074731242971705112228'+  comment: $R/\rho=\frac{2\sqrt{165}}{11}$; $R^2=\frac{30}{11}$ with $\mu=2$+- params:+    family: A+    n: '10'+    expression: density+  number: '116.0151277666199205534033600290105967016136957189555050705394796936231945156708878183957860599676542'+  comment: $\Theta(A_{10})=\frac{202500\sqrt{11}\,\pi^{5}}{1771561}$+- params:+    family: A+    n: '11'+    expression: radius+  number: '2.449489742783178098197284074705891391965947480656670128432692567250960377457315026539859433104640235'+  comment: $R/\rho=\sqrt{6}$; $R^2=3$ with $\mu=2$+- params:+    family: A+    n: '11'+    expression: density+  number: '228.9186213458728793238168760844401419904587104126898630633917500821705915181320494854692159386220110'+  comment: $\Theta(A_{11})=\frac{288\pi^{5}}{385}$+- params:+    family: A+    n: '12'+    expression: radius+  number: '2.541955637208970163356527125017880063513596897814620095448453248091796466218290498699055645452203161'+  comment: $R/\rho=\frac{2\sqrt{273}}{13}$; $R^2=\frac{42}{13}$ with $\mu=2$+- params:+    family: A+    n: '12'+    expression: density+  number: '421.1440721554849571039071732338507912431303908887815319863499991010193924434978791470135655291125746'+  comment: $\Theta(A_{12})=\frac{38118276\sqrt{13}\,\pi^{6}}{313742585}$+- params:+    family: A+    n: '13'+    expression: radius+  number: '2.645751311064590590501615753639260425710259183082450180368334459201068823230283627760392886474543611'+  comment: $R/\rho=\sqrt{7}$; $R^2=\frac{7}{2}$ with $\mu=2$+- params:+    family: A+    n: '13'+    expression: density+  number: '836.9887685273318660019113281256783859239225586354908300503324259541655733302968162797935365768582117'+  comment: $\Theta(A_{13})=\frac{16807\pi^{6}}{19305}$+- params:+    family: A+    n: '14'+    expression: radius+  number: '2.732520204255892902192276981472280530862675376921288781211194266637426129521427752780595100900945797'+  comment: $R/\rho=\frac{4\sqrt{105}}{15}$; $R^2=\frac{56}{15}$ with $\mu=2$+- params:+    family: A+    n: '14'+    expression: density+  number: '1564.048868967261339918402173297274792210321487414513307370756756514193122623198997939704705761844104'+  comment: $\Theta(A_{14})=\frac{15420489728\sqrt{15}\,\pi^{7}}{115330078125}$+- params:+    family: A+    n: '15'+    expression: radius+  number: '2.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145'+  comment: $R/\rho=2\sqrt{2}$; $R^2=4$ with $\mu=2$+- params:+    family: A+    n: '15'+    expression: density+  number: '3124.783356504510323242956142799925344724069683615497315592835990759993405993578146956173601913375561'+  comment: $\Theta(A_{15})=\frac{2097152\pi^{7}}{2027025}$+- params:+    family: A+    n: '16'+    expression: radius+  number: '2.910427500435995682226877545393466135398022982616673247810800169243497185650085830767903753513424209'+  comment: $R/\rho=\frac{12\sqrt{17}}{17}$; $R^2=\frac{72}{17}$ with $\mu=2$+- params:+    family: A+    n: '16'+    expression: density+  number: '5909.117119637672521178938156835114526317382294948169260067935599042092472828903412118638899475368048'+  comment: $\Theta(A_{16})=\frac{626913312768\sqrt{17}\,\pi^{8}}{4150575677395}$+- params:+    family: A+    n: '17'+    expression: radius+  number: '3'+  comment: $R/\rho=3$; $R^2=\frac{9}{2}$ with $\mu=2$+- params:+    family: A+    n: '17'+    expression: density+  number: '11853.07495260401227225108372655982963333735853696517277305677638059487132561125568582331362147325445'+  comment: $\Theta(A_{17})=\frac{531441\pi^{8}}{425425}$+- params:+    family: A+    n: '18'+    expression: radius+  number: '3.077935056255462286370025221020926410290289652738155062734236843911604770250605060057488030327810167'+  comment: $R/\rho=\frac{6\sqrt{95}}{19}$; $R^2=\frac{90}{19}$ with $\mu=2$+- params:+    family: A+    n: '18'+    expression: density+  number: '22626.07369227618881414884369100518160995594098808618999002915841013185819264619700550038521647485653'+  comment: $\Theta(A_{18})=\frac{7473389062500\sqrt{19}\,\pi^{9}}{42917463804607}$+- params:+    family: A+    n: '19'+    expression: radius+  number: '3.162277660168379331998893544432718533719555139325216826857504852792594438639238221344248108379300295'+  comment: $R/\rho=\sqrt{10}$; $R^2=5$ with $\mu=2$+- params:+    family: A+    n: '19'+    expression: density+  number: '45528.90725594584548808895099280516198836603945576825546760872965333731487930813552082556576637603755'+  comment: $\Theta(A_{19})=\frac{40000000\pi^{9}}{26189163}$+- params:+    family: A+    n: '20'+    expression: radius+  number: '3.236694374850748275461428822564184587358224216591466526958304990085313815593854740342786968592845840'+  comment: $R/\rho=\frac{2\sqrt{1155}}{21}$; $R^2=\frac{110}{21}$ with $\mu=2$+- params:+    family: A+    n: '20'+    expression: density+  number: '87570.92778545918962448858509365594528786886967788841927199463253952507186427440208213583374808965141'+  comment: $\Theta(A_{20})=\frac{40527225939062500\sqrt{21}\,\pi^{10}}{198607342807439307}$+- params:+    family: A+    n: '21'+    expression: radius+  number: '3.316624790355399849114932736670686683927088545589353597058682146116484642609043846708843399128290651'+  comment: $R/\rho=\sqrt{11}$; $R^2=\frac{11}{2}$ with $\mu=2$+- params:+    family: A+    n: '21'+    expression: density+  number: '176662.6157138625626923172658354633373826425908419485943852861451933889245441946127689478956095974321'+  comment: $\Theta(A_{21})=\frac{2357947691\pi^{10}}{1249937325}$+- params:+    family: A+    n: '22'+    expression: radius+  number: '3.387958215439679443642581910197069263383989530041460190338454563911336627786030601895796540773809870'+  comment: $R/\rho=\frac{2\sqrt{1518}}{23}$; $R^2=\frac{132}{23}$ with $\mu=2$+- params:+    family: A+    n: '22'+    expression: density+  number: '341929.0586685800579507269337962683253952744794417548676661526392367711712508086234450897711072557967'+  comment: $\Theta(A_{22})=\frac{929384818317508608\sqrt{23}\,\pi^{11}}{3835059275603556175}$+- params:+    family: A+    n: '23'+    expression: radius+  number: '3.464101615137754587054892683011744733885610507620761256111613958903866033817600074162292373514497151'+  comment: $R/\rho=2\sqrt{3}$; $R^2=6$ with $\mu=2$+- params:+    family: A+    n: '23'+    expression: density+  number: '691247.4592887740181169881248915556892839576444771018580762901603589617462279375674443649435381925856'+  comment: $\Theta(A_{23})=\frac{3057647616\pi^{11}}{1301375075}$+- params:+    family: A+    n: '24'+    expression: radius+  number: '3.532704346531138741905617090783701585669855792372567130484081189022855975294177136148174215765908607'+  comment: $R/\rho=\frac{2\sqrt{78}}{5}$; $R^2=\frac{156}{25}$ with $\mu=2$+- params:+    family: A+    n: '24'+    expression: density+  number: '1344951.365215002441106465210126379275553804007369829343396956807252656785760648054718974570468529901'+  comment: $\Theta(A_{24})=\frac{834812512876395675648\pi^{12}}{573694705963134765625}$+- params:+    family: D+    n: '4'+    expression: radius+  number: '1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573'+  comment: $R/\rho=\sqrt{2}$; $R^2=1$ with $\mu=2$; $D_4^{*}\cong D_4$+  equals: HREF{Algebraic_numbers_of_degree_2#1,0,-2,2}+- params:+    family: D+    n: '4'+    expression: density+  number: '2.467401100272339654708622749969037783828424851810197656603337344055011205604801310750443350929638058'+  comment: $\Theta(D_{4})=\frac{\pi^{2}}{4}$; $D_4^{*}\cong D_4$+- params:+    family: D+    n: '5'+    expression: radius+  number: '1.581138830084189665999446772216359266859777569662608413428752426396297219319619110672124054189650148'+  comment: $R/\rho=\frac{\sqrt{10}}{2}$; $R^2=\frac{5}{4}$ with $\mu=2$+  equals: HREF{Algebraic_numbers_of_degree_2#2,0,-5,2}+- params:+    family: D+    n: '5'+    expression: density+  number: '4.597730489972271939654968592204192888413062967792622812123160170349463570562389372938636500068838570'+  comment: $\Theta(D_{5})=\frac{5\sqrt{5}\,\pi^{2}}{24}$+- params:+    family: D+    n: '6'+    expression: radius+  number: '1.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576'+  comment: $R/\rho=\sqrt{3}$; $R^2=\frac{3}{2}$ with $\mu=2$+  equals: HREF{Algebraic_numbers_of_degree_2#1,0,-3,2}+- params:+    family: D+    n: '6'+    expression: density+  number: '8.720515316334324424352713612622267400625862409155186538978151341695548570537229823065634665419561742'+  comment: $\Theta(D_{6})=\frac{9\pi^{3}}{32}$+- params:+    family: D+    n: '7'+    expression: radius+  number: '1.870828693386970692791874366158274650878009903889363473151872733660017578153469513988404947597189786'+  comment: $R/\rho=\frac{\sqrt{14}}{2}$; $R^2=\frac{7}{4}$ with $\mu=2$+- params:+    family: D+    n: '7'+    expression: density+  number: '16.74879150720249937623341649164595376584047882981056036581488544812874667846202831366940731906962842'+  comment: $\Theta(D_{7})=\frac{49\sqrt{7}\,\pi^{3}}{240}$+- params:+    family: D+    n: '8'+    expression: radius+  number: '2'+  comment: $R/\rho=2$; $R^2=2$ with $\mu=2$+- params:+    family: D+    n: '8'+    expression: density+  number: '32.46969701133414574548011089623503708324252855756180723048928646332361848560906539670728907825099752'+  comment: $\Theta(D_{8})=\frac{\pi^{4}}{3}$+- params:+    family: D+    n: '9'+    expression: radius+  number: '2.121320343559642573202533086314547117854507813065422109765019606986098717693160558275581301491462359'+  comment: $R/\rho=\frac{3\sqrt{2}}{2}$; $R^2=\frac{9}{4}$ with $\mu=2$+- params:+    family: D+    n: '9'+    expression: density+  number: '63.40288157481051495121875225898752330451018746016756465453577633507924431073841609160610822690618890'+  comment: $\Theta(D_{9})=\frac{729\pi^{4}}{1120}$+- params:+    family: D+    n: '10'+    expression: radius+  number: '2.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275'+  comment: $R/\rho=\sqrt{5}$; $R^2=\frac{5}{2}$ with $\mu=2$+  equals: HREF{Algebraic_numbers_of_degree_2#1,0,-5,2}+- params:+    family: D+    n: '10'+    expression: density+  number: '124.5197285096360080007899210788719101889244411735331179009979018905578581272843524112074790338155594'+  comment: $\Theta(D_{10})=\frac{625\pi^{5}}{1536}$+- params:+    family: D+    n: '11'+    expression: radius+  number: '2.345207879911714777282815056772233140294114176705868576802850945508512316376619860741057798030771568'+  comment: $R/\rho=\frac{\sqrt{22}}{2}$; $R^2=\frac{11}{4}$ with $\mu=2$+- params:+    family: D+    n: '11'+    expression: density+  number: '245.6997214891637038652516366755274286839542530324780275688313198892091407223667749134798224660769724'+  comment: $\Theta(D_{11})=\frac{14641\sqrt{11}\,\pi^{5}}{60480}$+- params:+    family: D+    n: '12'+    expression: radius+  number: '2.449489742783178098197284074705891391965947480656670128432692567250960377457315026539859433104640235'+  comment: $R/\rho=\sqrt{6}$; $R^2=3$ with $\mu=2$+- params:+    family: D+    n: '12'+    expression: density+  number: '486.7032792474978712465485933490375738015288735910298493201698199138225433764397955839808383831573414'+  comment: $\Theta(D_{12})=\frac{81\pi^{6}}{160}$+- params:+    family: D+    n: '13'+    expression: radius+  number: '2.549509756796392415014112054511390994781885473049798203792485402212966816031112097794174425546966004'+  comment: $R/\rho=\frac{\sqrt{26}}{2}$; $R^2=\frac{13}{4}$ with $\mu=2$+- params:+    family: D+    n: '13'+    expression: density+  number: '967.2822325400784041861998603347598252633578565554118337372822829054336921706381920000638962698480395'+  comment: $\Theta(D_{13})=\frac{371293\sqrt{13}\,\pi^{6}}{1330560}$+- params:+    family: D+    n: '14'+    expression: radius+  number: '2.645751311064590590501615753639260425710259183082450180368334459201068823230283627760392886474543611'+  comment: $R/\rho=\sqrt{7}$; $R^2=\frac{7}{2}$ with $\mu=2$+- params:+    family: D+    n: '14'+    expression: density+  number: '1927.812922931379177251404512279908037561056929720208229225113573103208607163770313479427151985224367'+  comment: $\Theta(D_{14})=\frac{117649\pi^{7}}{184320}$+- params:+    family: D+    n: '15'+    expression: radius+  number: '2.738612787525830567284848914004010669763723474989916271134472248662466385613613669004292180819353129'+  comment: $R/\rho=\frac{\sqrt{30}}{2}$; $R^2=\frac{15}{4}$ with $\mu=2$+- params:+    family: D+    n: '15'+    expression: density+  number: '3851.540610753371660119237090711667214836194569918658352174187124324005311182715301583298012352092184'+  comment: $\Theta(D_{15})=\frac{84375\sqrt{15}\,\pi^{7}}{256256}$+- params:+    family: D+    n: '16'+    expression: radius+  number: '2.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145'+  comment: $R/\rho=2\sqrt{2}$; $R^2=4$ with $\mu=2$+- params:+    family: D+    n: '16'+    expression: density+  number: '7711.314095600212526428302631746450933188818872190848156127566356371837075774070894555805929259816738'+  comment: $\Theta(D_{16})=\frac{256\pi^{8}}{315}$+- params:+    family: D+    n: '17'+    expression: radius+  number: '2.915475947422650235437076438772791538260699167442985977225003372433905030998356313832620163226517699'+  comment: $R/\rho=\frac{\sqrt{34}}{2}$; $R^2=\frac{17}{4}$ with $\mu=2$+- params:+    family: D+    n: '17'+    expression: density+  number: '15468.09482832278955627140264900308815321696338382458716345567235102493863491392699045847774418313621'+  comment: $\Theta(D_{17})=\frac{410338673\sqrt{17}\,\pi^{8}}{1037836800}$+- params:+    family: D+    n: '18'+    expression: radius+  number: '3'+  comment: $R/\rho=3$; $R^2=\frac{9}{2}$ with $\mu=2$+- params:+    family: D+    n: '18'+    expression: density+  number: '31079.10113302915160442958502494917366595996103006234869066190937050775585189181861015068273980671484'+  comment: $\Theta(D_{18})=\frac{4782969\pi^{9}}{4587520}$+- params:+    family: D+    n: '19'+    expression: radius+  number: '3.082207001484488225125096190727122112617812011722287272437286036229199825131832106483364899599449973'+  comment: $R/\rho=\frac{\sqrt{38}}{2}$; $R^2=\frac{19}{4}$ with $\mu=2$+- params:+    family: D+    n: '19'+    expression: density+  number: '62538.35872402732263916206252461473703794965889527943245429211254706998861378310229967593275660390010'+  comment: $\Theta(D_{19})=\frac{16983563041\sqrt{19}\,\pi^{9}}{35286451200}$+- params:+    family: D+    n: '20'+    expression: radius+  number: '3.162277660168379331998893544432718533719555139325216826857504852792594438639238221344248108379300295'+  comment: $R/\rho=\sqrt{10}$; $R^2=5$ with $\mu=2$+- params:+    family: D+    n: '20'+    expression: density+  number: '126010.2118653030274052651086567326643417306691804037401943475220603852018356824016384215735021313726'+  comment: $\Theta(D_{20})=\frac{390625\pi^{10}}{290304}$+- params:+    family: D+    n: '21'+    expression: radius+  number: '3.240370349203930115482983718043998328852602153529173274855677198904808688922022185700180453302805118'+  comment: $R/\rho=\frac{\sqrt{42}}{2}$; $R^2=\frac{21}{4}$ with $\mu=2$+- params:+    family: D+    n: '21'+    expression: density+  number: '254208.7219781498555560951854170884174398226618680378193793482699273454753763321073707299213205108349'+  comment: $\Theta(D_{21})=\frac{1400846643\sqrt{21}\,\pi^{10}}{2364876800}$+- params:+    family: D+    n: '22'+    expression: radius+  number: '3.316624790355399849114932736670686683927088545589353597058682146116484642609043846708843399128290651'+  comment: $R/\rho=\sqrt{11}$; $R^2=\frac{11}{2}$ with $\mu=2$+- params:+    family: D+    n: '22'+    expression: density+  number: '513395.9878527280446916643806885169701221843324695595458265368406507334217321491427061587722598047905'+  comment: $\Theta(D_{22})=\frac{25937424601\pi^{11}}{14863564800}$+- params:+    family: D+    n: '23'+    expression: radius+  number: '3.391164991562634069532278163312984552597874161961644116375109791040364131993556294931320470468165022'+  comment: $R/\rho=\frac{\sqrt{46}}{2}$; $R^2=\frac{23}{4}$ with $\mu=2$+- params:+    family: D+    n: '23'+    expression: density+  number: '1037886.845868297340940238884306113158695240503868522001794415835242396945444433517339734965164644836'+  comment: $\Theta(D_{23})=\frac{41426511213649\sqrt{23}\,\pi^{11}}{56317176115200}$+- params:+    family: D+    n: '24'+    expression: radius+  number: '3.464101615137754587054892683011744733885610507620761256111613958903866033817600074162292373514497151'+  comment: $R/\rho=2\sqrt{3}$; $R^2=6$ with $\mu=2$+- params:+    family: D+    n: '24'+    expression: density+  number: '2100131.636354929189889642229167777821181486755503260085892983530295843576549330947766615138450928655'+  comment: $\Theta(D_{24})=\frac{4374\pi^{12}}{1925}$+- params:+    family: E+    n: '6'+    expression: radius+  number: '1.632993161855452065464856049803927594643964987104446752288461711500640251638210017693239622069760157'+  comment: $R/\rho=\frac{2\sqrt{6}}{3}$; $R^2=\frac{4}{3}$ with $\mu=2$+- params:+    family: E+    n: '6'+    expression: density+  number: '7.072190494000638853944463334968094461983569488941271478260699317992925336266585496130877091751642885'+  comment: $\Theta(E_{6})=\frac{32\sqrt{3}\,\pi^{3}}{243}$+- params:+    family: E+    n: '7'+    expression: radius+  number: '1.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576'+  comment: $R/\rho=\sqrt{3}$; $R^2=\frac{3}{2}$ with $\mu=2$+  equals: HREF{Algebraic_numbers_of_degree_2#1,0,-3,2}+- params:+    family: E+    n: '7'+    expression: density+  number: '13.80971483069589033355048331569216302531970220742371493121084768611654259858484150003413232112575267'+  comment: $\Theta(E_{7})=\frac{9\sqrt{3}\,\pi^{3}}{35}$+- params:+    family: E+    n: '8'+    expression: radius+  number: '1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573'+  comment: $R/\rho=\sqrt{2}$; $R^2=1$ with $\mu=2$; $E_8^{*}=E_8$+  equals: HREF{Algebraic_numbers_of_degree_2#1,0,-2,2}+- params:+    family: E+    n: '8'+    expression: density+  number: '4.058712126416768218185013862029379635405316069695225903811160807915452310701133174588411134781374690'+  comment: $\Theta(E_{8})=\frac{\pi^{4}}{24}$; not a locally thinnest lattice covering+    CITE{SVLeech}; $E_8^{*}=E_8$+- params:+    family: A*+    n: '3'+    expression: radius+  number: '1.290994448735805628393088466594133203610973901763863608862524588704494363978993011173095792286224506'+  comment: $R/\rho=\frac{\sqrt{15}}{3}$; $R^2=\frac{5}{16}$ with $\mu=\frac{3}{4}$;+    $A_3^{*}$ is the body-centred cubic lattice+  equals: HREF{Algebraic_numbers_of_degree_2#3,0,-5,2}+- params:+    family: A*+    n: '3'+    expression: density+  number: '1.463503068966817998574244717334353083222189413972333837497858547123362891795664209396696336534534842'+  comment: $\Theta(A_{3}^{*})=\frac{5\sqrt{5}\,\pi}{24}$; the thinnest lattice covering+    in dimension 3 CITE{Bambah}; $A_3^{*}$ is the body-centred cubic lattice+- params:+    family: A*+    n: '4'+    expression: radius+  number: '1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573'+  comment: $R/\rho=\sqrt{2}$; $R^2=\frac{2}{5}$ with $\mu=\frac{4}{5}$+  equals: HREF{Algebraic_numbers_of_degree_2#1,0,-2,2}+- params:+    family: A*+    n: '4'+    expression: density+  number: '1.765528508149352424827507939406410069150616179632367159855293505414194011095957519208436416026434011'+  comment: $\Theta(A_{4}^{*})=\frac{2\sqrt{5}\,\pi^{2}}{25}$; the thinnest lattice+    covering in dimension 4 CITE{DeloneRyshkov}+- params:+    family: A*+    n: '5'+    expression: radius+  number: '1.527525231651946668862682397909336162994818858922657300869080707968956141849256962201453853164481678'+  comment: $R/\rho=\frac{\sqrt{21}}{3}$; $R^2=\frac{35}{72}$ with $\mu=\frac{5}{6}$+- params:+    family: A*+    n: '5'+    expression: density+  number: '2.124285908991589721102606025324353057687042692239452369433234210002194459218544627794852930849704785'+  comment: $\Theta(A_{5}^{*})=\frac{245\sqrt{105}\,\pi^{2}}{11664}$; the thinnest+    lattice covering in dimension 5 CITE{RyshkovBaranovskii}+- params:+    family: A*+    n: '6'+    expression: radius+  number: '1.632993161855452065464856049803927594643964987104446752288461711500640251638210017693239622069760157'+  comment: $R/\rho=\frac{2\sqrt{6}}{3}$; $R^2=\frac{4}{7}$ with $\mu=\frac{6}{7}$+- params:+    family: A*+    n: '6'+    expression: density+  number: '2.551133828668911667945352901684633881153782697942228508150538867567655827742178406794412015042437601'+  comment: $\Theta(A_{6}^{*})=\frac{32\sqrt{7}\,\pi^{3}}{1029}$; a thinner lattice+    covering in dimension 6 is known CITE{SV}+- params:+    family: A*+    n: '7'+    expression: radius+  number: '1.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576'+  comment: $R/\rho=\sqrt{3}$; $R^2=\frac{21}{32}$ with $\mu=\frac{7}{8}$+  equals: HREF{Algebraic_numbers_of_degree_2#1,0,-3,2}+- params:+    family: A*+    n: '7'+    expression: density+  number: '3.059622898859479486587355087153620651585032899444023032172741646200423777967454039813256237342164178'+  comment: $\Theta(A_{7}^{*})=\frac{441\sqrt{21}\,\pi^{3}}{20480}$; a thinner lattice+    covering in dimension 7 is known CITE{SV}+- params:+    family: A*+    n: '8'+    expression: radius+  number: '1.825741858350553711523232609336007113175815649993277514089648165774977590409075779336194787212902086'+  comment: $R/\rho=\frac{\sqrt{30}}{3}$; $R^2=\frac{20}{27}$ with $\mu=\frac{8}{9}$+- params:+    family: A*+    n: '8'+    expression: density+  number: '3.665847800000468057091580540030035742433405991358793231665146625494489715954440707063319060243864984'+  comment: $\Theta(A_{8}^{*})=\frac{20000\pi^{4}}{531441}$; a thinner lattice covering+    in dimension 8 is known CITE{SV}+- params:+    family: A*+    n: '9'+    expression: radius+  number: '1.914854215512676219950203822739643106073421485994264122566625823521966907152468209009041788532262707'+  comment: $R/\rho=\frac{\sqrt{33}}{3}$; $R^2=\frac{33}{40}$ with $\mu=\frac{9}{10}$+- params:+    family: A*+    n: '9'+    expression: density+  number: '4.388947936151627216724177374225484490372238023249542626136580827006776588271738702298506165338957413'+  comment: $\Theta(A_{9}^{*})=\frac{43923\sqrt{33}\,\pi^{4}}{5600000}$; a thinner+    lattice covering in dimension 9 is known CITE{DSV}+- params:+    family: A*+    n: '10'+    expression: radius+  number: '2'+  comment: $R/\rho=2$; $R^2=\frac{10}{11}$ with $\mu=\frac{10}{11}$+- params:+    family: A*+    n: '10'+    expression: density+  number: '5.251713602604193934516201482794718369208850423491812986732239821522037611820492864207216652920346488'+  comment: $\Theta(A_{10}^{*})=\frac{2500\sqrt{11}\,\pi^{5}}{483153}$; a thinner lattice+    covering in dimension 10 is known CITE{DSV}+- params:+    family: A*+    n: '11'+    expression: radius+  number: '2.081665999466132735282297706979931487024319992663885436150990643536537066861171500211112037407113529'+  comment: $R/\rho=\frac{\sqrt{39}}{3}$; $R^2=\frac{143}{144}$ with $\mu=\frac{11}{12}$+- params:+    family: A*+    n: '11'+    expression: density+  number: '6.281306221903178204445738766734193692744792119654676997808894382475618320972943181572755816112586911'+  comment: $\Theta(A_{11}^{*})=\frac{5436100813\sqrt{429}\,\pi^{5}}{5485491486720}$;+    a thinner lattice covering in dimension 11 is known CITE{DSV}+- params:+    family: A*+    n: '12'+    expression: radius+  number: '2.160246899469286743655322478695998885901734769019448849903784799269872459281348123800120302201870079'+  comment: $R/\rho=\frac{\sqrt{42}}{3}$; $R^2=\frac{14}{13}$ with $\mu=\frac{12}{13}$+- params:+    family: A*+    n: '12'+    expression: density+  number: '7.510113769576549303636204735308724672374067327234787264502812055299385599129591809206003226170731784'+  comment: $\Theta(A_{12}^{*})=\frac{470596\sqrt{13}\,\pi^{6}}{217206405}$; a thinner+    lattice covering in dimension 12 is known CITE{DSV}+- params:+    family: A*+    n: '13'+    expression: radius+  number: '2.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275'+  comment: $R/\rho=\sqrt{5}$; $R^2=\frac{65}{56}$ with $\mu=\frac{13}{14}$+  equals: HREF{Algebraic_numbers_of_degree_2#1,0,-5,2}+- params:+    family: A*+    n: '13'+    expression: density+  number: '8.976768394939004134904601717190052647825931149143188591614851194586406664698145765595222254395956857'+  comment: $\Theta(A_{13}^{*})=\frac{1160290625\sqrt{65}\,\pi^{6}}{1001849942016}$;+    a thinner lattice covering in dimension 13 is known CITE{DSV}+- params:+    family: A*+    n: '14'+    expression: radius+  number: '2.309401076758503058036595122007829822590407005080507504074409305935910689211733382774861582342998101'+  comment: $R/\rho=\frac{4\sqrt{3}}{3}$; $R^2=\frac{56}{45}$ with $\mu=\frac{14}{15}$+- params:+    family: A*+    n: '14'+    expression: density+  number: '10.72735849771784183757477485114728938415858358994865094218626033274480879714128256474420237148041224'+  comment: $\Theta(A_{14}^{*})=\frac{15420489728\sqrt{15}\,\pi^{7}}{16815125390625}$;+    a thinner lattice covering in dimension 14 is known CITE{DSV}+- params:+    family: A*+    n: '15'+    expression: radius+  number: '2.380476142847616665999799937122421759588723719967577944402810294255273468813340628752996071177606806'+  comment: $R/\rho=\frac{\sqrt{51}}{3}$; $R^2=\frac{85}{64}$ with $\mu=\frac{15}{16}$+- params:+    family: A*+    n: '15'+    expression: density+  number: '12.81687351505401476768881764529366219416442403965739985163645692129908816829392641620599011438092602'+  comment: $\Theta(A_{15}^{*})=\frac{1282308353125\sqrt{85}\,\pi^{7}}{2785921946615808}$;+    a thinner lattice covering in dimension 15 is known CITE{DSV}+- params:+    family: A*+    n: '16'+    expression: radius+  number: '2.449489742783178098197284074705891391965947480656670128432692567250960377457315026539859433104640235'+  comment: $R/\rho=\sqrt{6}$; $R^2=\frac{24}{17}$ with $\mu=\frac{16}{17}$+- params:+    family: A*+    n: '16'+    expression: density+  number: '15.31092684557848389880230889593003306620873327451895708293779990606852187747162902088353319479976479'+  comment: $\Theta(A_{16}^{*})=\frac{95551488\sqrt{17}\,\pi^{8}}{244151510435}$; the+    thinnest lattice covering in dimension 16 listed by Dutour Sikirić, Schürmann+    and Vallentin CITE{DSV}+- params:+    family: A*+    n: '17'+    expression: radius+  number: '2.516611478423583232412228268982039019407398234874460046099542301903573984639270398527484024715841324'+  comment: $R/\rho=\frac{\sqrt{57}}{3}$; $R^2=\frac{323}{216}$ with $\mu=\frac{17}{18}$+- params:+    family: A*+    n: '17'+    expression: density+  number: '18.28781095600886607219344412662745149952538841637398685811818043343032577853446429618361313411570534'+  comment: $\Theta(A_{17}^{*})=\frac{6969012721055784593\sqrt{969}\,\pi^{8}}{112556454284898931507200}$;+    a thinner lattice covering in dimension 17 is known CITE{DSV}+- params:+    family: A*+    n: '18'+    expression: radius+  number: '2.581988897471611256786176933188266407221947803527727217725049177408988727957986022346191584572449012'+  comment: $R/\rho=\frac{2\sqrt{15}}{3}$; $R^2=\frac{30}{19}$ with $\mu=\frac{18}{19}$+- params:+    family: A*+    n: '18'+    expression: density+  number: '21.84094905010656848391139715130307628863297661807842350305105978725322896206257903289677991734096805'+  comment: $\Theta(A_{18}^{*})=\frac{379687500\sqrt{19}\,\pi^{9}}{2258813884453}$;+    the thinnest lattice covering in dimension 18 listed by Dutour Sikirić, Schürmann+    and Vallentin CITE{DSV}+- params:+    family: A*+    n: '19'+    expression: radius+  number: '2.645751311064590590501615753639260425710259183082450180368334459201068823230283627760392886474543611'+  comment: $R/\rho=\sqrt{7}$; $R^2=\frac{133}{80}$ with $\mu=\frac{19}{20}$+- params:+    family: A*+    n: '19'+    expression: density+  number: '26.08182004558247299120570192465693329051955846418696703022882245563458643273853955842322756512379959'+  comment: $\Theta(A_{19}^{*})=\frac{97906861202319841\sqrt{133}\,\pi^{9}}{1290475929600000000000}$;+    a thinner lattice covering in dimension 19 is known CITE{DSV}+- params:+    family: A*+    n: '20'+    expression: radius+  number: '2.708012801545320120153294522755346782834734695575104275883858531510972267613557564999719200982645626'+  comment: $R/\rho=\frac{\sqrt{66}}{3}$; $R^2=\frac{110}{63}$ with $\mu=\frac{20}{21}$+- params:+    family: A*+    n: '20'+    expression: density+  number: '31.14344838176163833619977115559577386653874347128074657846682049365825855052181144007269401192031499'+  comment: $\Theta(A_{20}^{*})=\frac{40527225939062500\sqrt{21}\,\pi^{10}}{558455475496975411383}$;+    a thinner lattice covering in dimension 20 is known CITE{DSV}+- params:+    family: A*+    n: '21'+    expression: radius+  number: '2.768874620972691617528087581635830670077404749651885259810729456345861677807717191289132836077452350'+  comment: $R/\rho=\frac{\sqrt{69}}{3}$; $R^2=\frac{161}{88}$ with $\mu=\frac{21}{22}$+- params:+    family: A*+    n: '21'+    expression: density+  number: '37.18456782695306450528575888269467295961046865148534255944523223839591639905882526228820219248192479'+  comment: $\Theta(A_{21}^{*})=\frac{238815593270954968849\sqrt{161}\,\pi^{10}}{7631529844136427572428800}$;+    a thinner lattice covering in dimension 21 is known CITE{DSV}+- params:+    family: A*+    n: '22'+    expression: radius+  number: '2.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145'+  comment: $R/\rho=2\sqrt{2}$; $R^2=\frac{44}{23}$ with $\mu=\frac{22}{23}$+- params:+    family: A*+    n: '22'+    expression: density+  number: '44.39458951818174359637317864436976908494816749456870258215781640358423760362071239838701606838887097'+  comment: $\Theta(A_{22}^{*})=\frac{424958764662784\sqrt{23}\,\pi^{11}}{13506078318429915225}$;+    a thinner lattice covering in dimension 22 is known CITE{DSV}+- params:+    family: A*+    n: '23'+    expression: radius+  number: '2.886751345948128822545743902509787278238008756350634380093011632419888361514666728468576977928747626'+  comment: $R/\rho=\frac{5\sqrt{3}}{3}$; $R^2=\frac{575}{288}$ with $\mu=\frac{23}{24}$+- params:+    family: A*+    n: '23'+    expression: density+  number: '52.99952960374942460718670124536522878256088982965332566626670899771059990599080305978934662983591161'+  comment: $\Theta(A_{23}^{*})=\frac{19753699881386280059814453125\sqrt{69}\,\pi^{11}}{910855701610617424251753685057536}$;+    a thinner lattice covering in dimension 23 is known CITE{DSV}+- params:+    family: A*+    n: '24'+    expression: radius+  number: '2.943920288775948951588014242319751321391546493643805942070067657519046646078480946790145179804923839'+  comment: $R/\rho=\frac{\sqrt{78}}{3}$; $R^2=\frac{52}{25}$ with $\mu=\frac{24}{25}$+- params:+    family: A*+    n: '24'+    expression: density+  number: '63.26908185551182732920800286985663862751481384433217155794137106718604632314067105845120015526323247'+  comment: $\Theta(A_{24}^{*})=\frac{381715826646728704\pi^{12}}{5576312541961669921875}$;+    the Leech lattice $\Lambda_{24}$ is a thinner lattice covering CITE{DSV}+- params:+    family: D*+    n: '5'+    expression: radius+  number: 3/2+  comment: $R/\rho=\frac{3}{2}$; $R^2=\frac{9}{16}$ with $\mu=1$+- params:+    family: D*+    n: '5'+    expression: density+  number: '2.498243614025743900392480534343650756126280162457825127310879060855698845674861327134823892816258534'+  comment: $\Theta(D_{5}^{*})=\frac{81\pi^{2}}{320}$+- params:+    family: D*+    n: '6'+    expression: radius+  number: '1.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576'+  comment: $R/\rho=\sqrt{3}$; $R^2=\frac{3}{4}$ with $\mu=1$+  equals: HREF{Algebraic_numbers_of_degree_2#1,0,-3,2}+- params:+    family: D*+    n: '6'+    expression: density+  number: '4.360257658167162212176356806311133700312931204577593269489075670847774285268614911532817332709780871'+  comment: $\Theta(D_{6}^{*})=\frac{9\pi^{3}}{64}$+- params:+    family: D*+    n: '7'+    expression: radius+  number: '1.802775637731994646559610633735247973125648286922623106355226528113583474146505222602309541009245359'+  comment: $R/\rho=\frac{\sqrt{13}}{2}$; $R^2=\frac{13}{16}$ with $\mu=1$+- params:+    family: D*+    n: '7'+    expression: density+  number: '4.568694211108658270461274542464035977395587936863368093527406036754464973587660675764016284625116151'+  comment: $\Theta(D_{7}^{*})=\frac{2197\sqrt{13}\,\pi^{3}}{53760}$+- params:+    family: D*+    n: '8'+    expression: radius+  number: '2'+  comment: $R/\rho=2$; $R^2=1$ with $\mu=1$+- params:+    family: D*+    n: '8'+    expression: density+  number: '8.117424252833536436370027724058759270810632139390451807622321615830904621402266349176822269562749379'+  comment: $\Theta(D_{8}^{*})=\frac{\pi^{4}}{12}$+- params:+    family: D*+    n: '9'+    expression: radius+  number: '2.061552812808830274910704927987038512573599612686810217199316786547477173168810796793931825405342148'+  comment: $R/\rho=\frac{\sqrt{17}}{2}$; $R^2=\frac{17}{16}$ with $\mu=1$+- params:+    family: D*+    n: '9'+    expression: density+  number: '8.666183496864602951956953728581011447868407247910625205443812669853112064185933898873460151579225058'+  comment: $\Theta(D_{9}^{*})=\frac{83521\sqrt{17}\,\pi^{4}}{3870720}$+- params:+    family: D*+    n: '10'+    expression: radius+  number: '2.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275'+  comment: $R/\rho=\sqrt{5}$; $R^2=\frac{5}{4}$ with $\mu=1$+  equals: HREF{Algebraic_numbers_of_degree_2#1,0,-5,2}+- params:+    family: D*+    n: '10'+    expression: density+  number: '15.56496606370450100009874013485898877361555514669163973762473773631973226591054405140093487922694493'+  comment: $\Theta(D_{10}^{*})=\frac{625\pi^{5}}{12288}$+- params:+    family: D*+    n: '11'+    expression: radius+  number: '2.291287847477920003294023596864004244492228288383985951303621061953434212773885443302180779746722516'+  comment: $R/\rho=\frac{\sqrt{21}}{2}$; $R^2=\frac{21}{16}$ with $\mu=1$+- params:+    family: D*+    n: '11'+    expression: density+  number: '16.81438765664326948399572725958410291735661041459349226719451898728610243097161146497441592488378607'+  comment: $\Theta(D_{11}^{*})=\frac{21609\sqrt{21}\,\pi^{5}}{1802240}$+- params:+    family: D*+    n: '12'+    expression: radius+  number: '2.449489742783178098197284074705891391965947480656670128432692567250960377457315026539859433104640235'+  comment: $R/\rho=\sqrt{6}$; $R^2=\frac{3}{2}$ with $\mu=1$+- params:+    family: D*+    n: '12'+    expression: density+  number: '30.41895495296861695290928708431484836259555459943936558251061374461390896102748722399880239894733384'+  comment: $\Theta(D_{12}^{*})=\frac{81\pi^{6}}{2560}$+- params:+    family: D*+    n: '13'+    expression: radius+  number: 5/2+  comment: $R/\rho=\frac{5}{2}$; $R^2=\frac{25}{16}$ with $\mu=1$+- params:+    family: D*+    n: '13'+    expression: density+  number: '33.12848111029900414134935618485423494982948311926463455542741825336278393102503355236808612898849115'+  comment: $\Theta(D_{13}^{*})=\frac{244140625\pi^{6}}{7084965888}$+- params:+    family: D*+    n: '14'+    expression: radius+  number: '2.645751311064590590501615753639260425710259183082450180368334459201068823230283627760392886474543611'+  comment: $R/\rho=\sqrt{7}$; $R^2=\frac{7}{4}$ with $\mu=1$+- params:+    family: D*+    n: '14'+    expression: density+  number: '60.24415384160559928910639100874712617378302905375650716328479915947526897386782229623209849953826148'+  comment: $\Theta(D_{14}^{*})=\frac{117649\pi^{7}}{5898240}$+- params:+    family: D*+    n: '15'+    expression: radius+  number: '2.692582403567252015625355245770164778147560080822394418840194335008322981413829346438316890839917742'+  comment: $R/\rho=\frac{\sqrt{29}}{2}$; $R^2=\frac{29}{16}$ with $\mu=1$+- params:+    family: D*+    n: '15'+    expression: density+  number: '66.00017379164128675001190474046585794739681999537693375619032686007967782185855351027627905501274891'+  comment: $\Theta(D_{15}^{*})=\frac{17249876309\sqrt{29}\,\pi^{7}}{4250979532800}$+- params:+    family: D*+    n: '16'+    expression: radius+  number: '2.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145'+  comment: $R/\rho=2\sqrt{2}$; $R^2=2$ with $\mu=1$+- params:+    family: D*+    n: '16'+    expression: density+  number: '120.4892827437533207254422286210382958310752948779820024394932243183099543089698577274344676446846365'+  comment: $\Theta(D_{16}^{*})=\frac{4\pi^{8}}{315}$+- params:+    family: D*+    n: '17'+    expression: radius+  number: '2.872281323269014329925305734109464659110132228991396183849938735282950360728702313513562682798394061'+  comment: $R/\rho=\frac{\sqrt{33}}{2}$; $R^2=\frac{33}{16}$ with $\mu=1$+- params:+    family: D*+    n: '17'+    expression: density+  number: '132.5987860734753264674647135793810769032044662970110755899386017708173024920346027870024405243423392'+  comment: $\Theta(D_{17}^{*})=\frac{1578460851\sqrt{33}\,\pi^{8}}{648858828800}$+- params:+    family: D*+    n: '18'+    expression: radius+  number: '3'+  comment: $R/\rho=3$; $R^2=\frac{9}{4}$ with $\mu=1$+- params:+    family: D*+    n: '18'+    expression: density+  number: '242.8054776017902469096061330074154192653121955473620991457961669570918425929048328918022089047399597'+  comment: $\Theta(D_{18}^{*})=\frac{4782969\pi^{9}}{587202560}$+- params:+    family: D*+    n: '19'+    expression: radius+  number: '3.041381265149109844499842122601033531042485047393205593209576523243166362659455119901533213978924332'+  comment: $R/\rho=\frac{\sqrt{37}}{2}$; $R^2=\frac{37}{16}$ with $\mu=1$+- params:+    family: D*+    n: '19'+    expression: density+  number: '268.1598305964027984638471678342584620723772999172741870372569777982262612334257923046849407128604662'+  comment: $\Theta(D_{19}^{*})=\frac{129961739795077\sqrt{37}\,\pi^{9}}{87876248902041600}$+- params:+    family: D*+    n: '20'+    expression: radius+  number: '3.162277660168379331998893544432718533719555139325216826857504852792594438639238221344248108379300295'+  comment: $R/\rho=\sqrt{10}$; $R^2=\frac{5}{2}$ with $\mu=1$+- params:+    family: D*+    n: '20'+    expression: density+  number: '492.2273900988399508018168306903619700848854264859521101341700080483796946706343814000842714927006742'+  comment: $\Theta(D_{20}^{*})=\frac{390625\pi^{10}}{74317824}$+- params:+    family: D*+    n: '21'+    expression: radius+  number: '3.201562118716424343244108837310906632260210066310509442764636313334091379098438037144677151124934982'+  comment: $R/\rho=\frac{\sqrt{41}}{2}$; $R^2=\frac{41}{16}$ with $\mu=1$+- params:+    family: D*+    n: '21'+    expression: density+  number: '545.1906894082681901536000204863069502751248347526646049925881541063922970915301692187996890996322921'+  comment: $\Theta(D_{21}^{*})=\frac{13422659310152401\sqrt{41}\,\pi^{10}}{14763209815542988800}$+- params:+    family: D*+    n: '22'+    expression: radius+  number: '3.316624790355399849114932736670686683927088545589353597058682146116484642609043846708843399128290651'+  comment: $R/\rho=\sqrt{11}$; $R^2=\frac{11}{4}$ with $\mu=1$+- params:+    family: D*+    n: '22'+    expression: density+  number: '1002.726538774859462288406993532259707269891274354608487942454766895963714320603794347966352069931232'+  comment: $\Theta(D_{22}^{*})=\frac{25937424601\pi^{11}}{7610145177600}$+- params:+    family: D*+    n: '23'+    expression: radius+  number: '3.354101966249684544613760503096914353160927539417288586406345868115781388456707349121621612568173412'+  comment: $R/\rho=\frac{3\sqrt{5}}{2}$; $R^2=\frac{45}{16}$ with $\mu=1$+- params:+    family: D*+    n: '23'+    expression: density+  number: '1113.253048406610343788994768380183759750388160034924922992343454564108130521569127272261428047522681'+  comment: $\Theta(D_{23}^{*})=\frac{756680642578125\sqrt{5}\,\pi^{11}}{447149070956363776}$+- params:+    family: D*+    n: '24'+    expression: radius+  number: '3.464101615137754587054892683011744733885610507620761256111613958903866033817600074162292373514497151'+  comment: $R/\rho=2\sqrt{3}$; $R^2=3$ with $\mu=1$+- params:+    family: D*+    n: '24'+    expression: density+  number: '2050.909801127860537001603739421658028497545659671152427629866728804534742723956003678335096143485015'+  comment: $\Theta(D_{24}^{*})=\frac{2187\pi^{12}}{985600}$+- params:+    family: E*+    n: '6'+    expression: radius+  number: '1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573'+  comment: $R/\rho=\sqrt{2}$; $R^2=\frac{2}{3}$ with $\mu=\frac{4}{3}$+  equals: HREF{Algebraic_numbers_of_degree_2#1,0,-2,2}+- params:+    family: E*+    n: '6'+    expression: density+  number: '2.652071435250239570229173750613035423243838558352976804347762244247347001099969561049078909406866082'+  comment: $\Theta(E_{6}^{*})=\frac{4\sqrt{3}\,\pi^{3}}{81}$+- params:+    family: E*+    n: '7'+    expression: radius+  number: '1.527525231651946668862682397909336162994818858922657300869080707968956141849256962201453853164481678'+  comment: $R/\rho=\frac{\sqrt{21}}{3}$; $R^2=\frac{7}{8}$ with $\mu=\frac{3}{2}$+- params:+    family: E*+    n: '7'+    expression: density+  number: '4.187197876800624844058354122911488441460119707452640091453721362032186669615507078417351829767407105'+  comment: $\Theta(E_{7}^{*})=\frac{49\sqrt{7}\,\pi^{3}}{960}$+- params:+    family: Lambda+    n: '24'+    expression: radius+  number: '1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573'+  comment: $R/\rho=\sqrt{2}$; $R^2=2$ with $\mu=4$; $\Lambda_{24}$ is the Leech lattice+  equals: HREF{Algebraic_numbers_of_degree_2#1,0,-2,2}+- params:+    family: Lambda+    n: '24'+    expression: density+  number: '7.903536371318468804212103428857682494130060554241242530753116640589806117892036736972176171770445469'+  comment: $\Theta(\Lambda_{24})=\frac{4\pi^{12}}{467775}$; a locally thinnest lattice+    covering CITE{SVLeech}; $\Lambda_{24}$ is the Leech lattice