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Packing densities and Hermite numbers of the classical lattices
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Numbers
family
$n$ 
$\Delta(L)$, $\delta(L)$ or $\gamma(L)$
A
1
centre:
1/2
comment: $\delta(A_{1})=\frac{1}{2}$; $\det=2$, $\mu=2$, kissing number $2$; $A_1=\sqrt{2}\,\mathbb{Z}$
A
1
density:
1
comment: $\Delta(A_{1})=1$; the densest packing of any kind in dimension 1; $A_1=\sqrt{2}\,\mathbb{Z}$
equals: One
A
1
hermite:
1
comment: $\gamma(A_{1})=1$; Hermite's constant $\gamma_{1}$; $A_1=\sqrt{2}\,\mathbb{Z}$
equals: One
A
10
centre:
0.009422229518055113207712877092814450806611047004515209082553074278740013189230238200877396020250825713
comment: $\delta(A_{10})=\frac{\sqrt{11}}{352}$; $\det=11$, $\mu=2$, kissing number $110$
A
10
density:
0.02402823089241500106101615809674920988862086845323238560585835100840754771083244875776808094231462279
comment: $\Delta(A_{10})=\frac{\sqrt{11}\,\pi^{5}}{42240}$
A
10
hermite:
1.573586884393544470910577389703060847050289960034329483222479690112842968652442417915296777326848230
comment: $\gamma(A_{10})=\frac{2}{11^{1/10}}$
A
11
centre:
0.006378879538497859630722093944546592166577988230876745126126803560549375982961757881614217273710000612
comment: $\delta(A_{11})=\frac{\sqrt{6}}{384}$; $\det=12$, $\mu=2$, kissing number $132$
A
11
density:
0.01201847168464467383503297013093922603134950303996848600699281026471812948055207148382095348220641627
comment: $\Delta(A_{11})=\frac{\sqrt{6}\,\pi^{5}}{62370}$
A
11
hermite:
1.595594790350010671043327255349365001093923225034600716432474987935317816928421338613405013027077721
comment: $\gamma(A_{11})=\frac{2^{9/11}}{3^{1/11}}$
A
12
centre:
0.004333595283009602515768294792632807627705885305102459390276986846426883351313714477409397935118378266
comment: $\delta(A_{12})=\frac{\sqrt{13}}{832}$; $\det=13$, $\mu=2$, kissing number $156$
A
12
density:
0.005786488436686590234019311923709057050834093105411363221367165329909345462276874389270809142187284009
comment: $\Delta(A_{12})=\frac{\sqrt{13}\,\pi^{6}}{599040}$
A
12
hermite:
1.615107310781479943692032140716542329654195974804270993044820930949102699395594766266221536198739415
comment: $\gamma(A_{12})=\frac{2}{13^{1/12}}$
A
13
centre:
0.002952847445384587712613410439329531725123057123975948862018230423215478597355227263125438489368910280
comment: $\delta(A_{13})=\frac{\sqrt{7}}{896}$; $\det=14$, $\mu=2$, kissing number $182$
A
13
density:
0.002688947792255565796332472953030458916732744761264711054285497493156665660148928078704263168035576243
comment: $\Delta(A_{13})=\frac{\sqrt{7}\,\pi^{6}}{945945}$
A
13
hermite:
1.632549283341321630287408419015642589110934389637549360001422436946971075177029699991942659693631158
comment: $\gamma(A_{13})=\frac{2^{12/13}}{7^{1/13}}$
A
14
centre:
0.002017178826149696294364200729053333130642146721506036888847694669850772443717176579957962175447225791
comment: $\delta(A_{14})=\frac{\sqrt{15}}{1920}$; $\det=15$, $\mu=2$, kissing number $210$
A
14
density:
0.001208823719808465618429741353024457419416352973384621441670032941077273546225795968453029609363729344
comment: $\Delta(A_{14})=\frac{\sqrt{15}\,\pi^{7}}{9676800}$
A
14
hermite:
1.648251490557794949911981416796352816339300520838035539580595781886398192656062395453211248896066979
comment: $\gamma(A_{14})=\frac{2}{3^{1/14}\cdot 5^{1/14}}$
A
15
centre:
0.001381067932004975633595399144736033279853195190797800852711601306631574685998151405127331576491837473
comment: $\delta(A_{15})=\frac{\sqrt{2}}{1024}$; $\det=16$, $\mu=2$, kissing number $240$
A
15
density:
0.0005267990830238343014942344870440350007947798198889932986102095778021866547206442444659482856140843291
comment: $\Delta(A_{15})=\frac{\sqrt{2}\,\pi^{7}}{8108100}$
A
15
hermite:
1.662475792285575556085280683775664373100417273320547235153185782890535235340516306696443430280054256
comment: $\gamma(A_{15})=2^{11/15}$
A
16
centre:
0.0009474047852981756778082283676411022576165439396538649895217448467589509067871373147030936697634844432
comment: $\delta(A_{16})=\frac{\sqrt{17}}{4352}$; $\det=17$, $\mu=2$, kissing number $272$
A
16
density:
0.0002229533653292515595032608571780731988836775933466674830516920285674050900256010281864321426646726454
comment: $\Delta(A_{16})=\frac{\sqrt{17}\,\pi^{8}}{175472640}$
A
16
hermite:
1.675432509473520540573916523910756553754036977519787999855545909138671812239108332465310135146355849
comment: $\gamma(A_{16})=\frac{2}{17^{1/16}}$
A
17
centre:
1/1536
comment: $\delta(A_{17})=\frac{1}{1536}$; $\det=18$, $\mu=2$, kissing number $306$
A
17
density:
0.00009178457481584572781010880191129873077004989173635449703634628663582274807575746737924834136590995668
comment: $\Delta(A_{17})=\frac{\pi^{8}}{103378275}$
A
17
hermite:
1.687292734883595117432729236880446158088124119992776131174272823042824383509922794626688643876976786
comment: $\gamma(A_{17})=\frac{2^{16/17}}{3^{2/17}}$
A
18
centre:
0.0004480776052159409490375187072224111491711558311299799482823133365706781792971996387393255711196489028
comment: $\delta(A_{18})=\frac{\sqrt{19}}{9728}$; $\det=19$, $\mu=2$, kissing number $342$
A
18
density:
0.00003680773215105456382239482196898414195518801818437606034613107298917086428366134564807742509518674705
comment: $\Delta(A_{18})=\frac{\sqrt{19}\,\pi^{9}}{3530096640}$
A
18
hermite:
1.698197233107663644258114840959473123346329995662803593305588173564765286009819641957041149154603362
comment: $\gamma(A_{18})=\frac{2}{19^{1/18}}$
A
19
centre:
0.0003088161777508182941405169476985076693085503065747282057478032082805268006483631075531492293339160445
comment: $\delta(A_{19})=\frac{\sqrt{10}}{10240}$; $\det=20$, $\mu=2$, kissing number $380$
A
19
density:
0.00001439750463073555763471912245959366331864715084979751906909650899098828277887365377359187405187304647
comment: $\Delta(A_{19})=\frac{\sqrt{10}\,\pi^{9}}{6547290750}$
A
19
hermite:
1.708262993375513107747813508738544085881574715989018696495543082141575575582458905876660260792716515
comment: $\gamma(A_{19})=\frac{2^{17/19}}{5^{1/19}}$
A
2
centre:
0.2886751345948128822545743902509787278238008756350634380093011632419888361514666728468576977928747626
comment: $\delta(A_{2})=\frac{\sqrt{3}}{6}$; $\det=3$, $\mu=2$, kissing number $6$; $A_{2}=\Lambda_{2}$
A
2
density:
0.9068996821171089252970391288210778661420331240463702877849424676940615905631769418420624941060300844
comment: $\Delta(A_{2})=\frac{\sqrt{3}\,\pi}{6}$; the densest packing of any kind in dimension 2 [13]; $A_{2}=\Lambda_{2}$
A
2
hermite:
1.154700538379251529018297561003914911295203502540253752037204652967955344605866691387430791171499050
comment: $\gamma(A_{2})=\frac{2}{\sqrt{3}}$; Hermite's constant $\gamma_{2}$ [9]; $A_{2}=\Lambda_{2}$
equals: Algebraic_numbers_of_degree_2#3,0,-4,2
A
20
centre:
0.0002131034084335863098301733256011908709535182559880939314828516612679905331820950003071224683544198769
comment: $\delta(A_{20})=\frac{\sqrt{21}}{21504}$; $\det=21$, $\mu=2$, kissing number $420$
A
20
density:
0.000005499536516287368162585488027923258154810059003100507260256746508041106132330487791988321874212521457
comment: $\Delta(A_{20})=\frac{\sqrt{21}\,\pi^{10}}{78033715200}$
A
20
hermite:
1.717588133299478501121941496385791503870438068950841220941884937760204197578598837351277574516192101
comment: $\gamma(A_{20})=\frac{2}{3^{1/20}\cdot 7^{1/20}}$
A
21
centre:
0.0001472223362196111438705137045752257938532976094455501419148917856053127060817224718887093128164191518
comment: $\delta(A_{21})=\frac{\sqrt{11}}{22528}$; $\det=22$, $\mu=2$, kissing number $462$
A
21
density:
0.000002053626511494816144477693541305358043241149208237410748849077115300199226698615359870862561598339641
comment: $\Delta(A_{21})=\frac{\sqrt{11}\,\pi^{10}}{151242416325}$
A
21
hermite:
1.726255622473788359715165126997843588765328008490425107640098302638643331340785087011837687400754226
comment: $\gamma(A_{21})=\frac{2^{20/21}}{11^{1/21}}$
A
22
centre:
0.0001018136787388060364639401762942147995923213961002065503245012974364864392813235411534517545524142081
comment: $\delta(A_{22})=\frac{\sqrt{23}}{47104}$; $\det=23$, $\mu=2$, kissing number $506$
A
22
density:
7.504106884735152640492964258639967853758404908248139824753027760630275003076909705640907794684904389e-7
comment: $\Delta(A_{22})=\frac{\sqrt{23}\,\pi^{11}}{1880240947200}$
A
22
hermite:
1.734336147394595049711209879648012028284610026953493822420039353140156496077575847807467040691158054
comment: $\gamma(A_{22})=\frac{2}{23^{1/22}}$
A
23
centre:
0.00007047732778193673883168320074486785347260763565309165967023954180712618070104166817550236762521356509
comment: $\delta(A_{23})=\frac{\sqrt{3}}{24576}$; $\det=24$, $\mu=2$, kissing number $552$
A
23
density:
2.685648792405079381122957708162599715143941715032483899447864135464203324732931460809555213349874405e-7
comment: $\Delta(A_{23})=\frac{\sqrt{3}\,\pi^{11}}{1897404859350}$
A
23
hermite:
1.741890342175930340199256595687853411363891921062025217911179240798942327455776880768285594628532936
comment: $\gamma(A_{23})=\frac{2^{20/23}}{3^{1/23}}$
A
24
centre:
1/20480
comment: $\delta(A_{24})=\frac{1}{20480}$; $\det=25$, $\mu=2$, kissing number $600$
A
24
density:
9.421749557636343007340554510185340039885116284181168712083240319001443526616140290465564932549530827e-8
comment: $\Delta(A_{24})=\frac{\pi^{12}}{9809952768000}$
A
24
hermite:
1.748970544442335671271718425882761979326529471699088254127749773872966802758875884305067209164683923
comment: $\gamma(A_{24})=\frac{2}{5^{1/12}}$
A
3
centre:
0.1767766952966368811002110905262122598212089844221185091470849672488415598077633798562984417909551966
comment: $\delta(A_{3})=\frac{\sqrt{2}}{8}$; $\det=4$, $\mu=2$, kissing number $12$; $A_{3}=\Lambda_{3}$; $A_3=D_3$, the face-centred cubic lattice
A
3
density:
0.7404804896930610411693134983434489497691036148959483705142326011594057988499123184292211557941275396
comment: $\Delta(A_{3})=\frac{\sqrt{2}\,\pi}{6}$; the densest packing of any kind in dimension 3 [7]; $A_{3}=\Lambda_{3}$; $A_3=D_3$, the face-centred cubic lattice
A
3
hermite:
1.259921049894873164767210607278228350570251464701507980081975112155299676513959483729396562436255094
comment: $\gamma(A_{3})=2^{1/3}$; Hermite's constant $\gamma_{3}$ [6]; $A_{3}=\Lambda_{3}$; $A_3=D_3$, the face-centred cubic lattice
A
4
centre:
0.1118033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137
comment: $\delta(A_{4})=\frac{\sqrt{5}}{20}$; $\det=5$, $\mu=2$, kissing number $20$
A
4
density:
0.5517276587966726327585962310645031466095675561351147374547792204419356284674867247526363800082606285
comment: $\Delta(A_{4})=\frac{\sqrt{5}\,\pi^{2}}{40}$
A
4
hermite:
1.337480609952844048006466146517295872776070383304955129539966906297538647242720943449191470376352353
comment: $\gamma(A_{4})=\frac{2}{5^{1/4}}$
A
5
centre:
0.07216878364870322056364359756274468195595021890876585950232529081049720903786666821171442444821869065
comment: $\delta(A_{5})=\frac{\sqrt{3}}{24}$; $\det=6$, $\mu=2$, kissing number $30$
A
5
density:
0.3798812505176037581053993966982966063494607803638771779523670617473220782275171526904027213139341401
comment: $\Delta(A_{5})=\frac{\sqrt{3}\,\pi^{2}}{45}$
A
5
hermite:
1.397654237543158490465954205542839687013584633139712792584031713336178783783983105452602535006347153
comment: $\gamma(A_{5})=\frac{2^{4/5}}{3^{1/5}}$
A
6
centre:
0.04724555912615340340181456702927250760196891398361518179229168677144765755768363621000701582990256448
comment: $\delta(A_{6})=\frac{\sqrt{7}}{56}$; $\det=7$, $\mu=2$, kissing number $42$
A
6
density:
0.2441514796968294369713326019190372269072956097640023376940945400601858116393881678377464623771082860
comment: $\Delta(A_{6})=\frac{\sqrt{7}\,\pi^{3}}{336}$
A
6
hermite:
1.446040052798967552509685118554498177089689708920244836572156252251105069335891595768411780348917303
comment: $\gamma(A_{6})=\frac{2}{7^{1/6}}$
A
7
centre:
1/32
comment: $\delta(A_{7})=\frac{1}{32}$; $\det=8$, $\mu=2$, kissing number $56$
A
7
density:
0.1476489365728562865498872146052447390582156598375481318768787528752685472260271716074604811076327914
comment: $\Delta(A_{7})=\frac{\pi^{3}}{210}$
A
7
hermite:
1.485994289136948424799853286714592606323711359437109733356503197150790858340556200496397994091233511
comment: $\gamma(A_{7})=2^{4/7}$
A
8
centre:
1/48
comment: $\delta(A_{8})=\frac{1}{48}$; $\det=9$, $\mu=2$, kissing number $72$
A
8
density:
0.08455650263368267121218778879227874240427741811865053966273251683157192313960694113725856530794530603
comment: $\Delta(A_{8})=\frac{\pi^{4}}{1152}$
A
8
hermite:
1.519671371303185094662375501309090670793546897774620637222577307400644466342094543188821228330074314
comment: $\gamma(A_{8})=\frac{2}{3^{1/4}}$
A
9
centre:
0.01397542485937368560255733542957047647150386474757203577669310778381575578523628062134009005236738922
comment: $\delta(A_{9})=\frac{\sqrt{5}}{160}$; $\det=10$, $\mu=2$, kissing number $90$
A
9
density:
0.04609806331819994245073740620365313147351893372623936529917451763099279502435109726330283538937166342
comment: $\Delta(A_{9})=\frac{\sqrt{5}\,\pi^{4}}{4725}$
A
9
hermite:
1.548527365362254119453358903073938556428667487479489470582783954818172283985407781713774776894218079
comment: $\gamma(A_{9})=\frac{2^{8/9}}{5^{1/9}}$
A*
10
centre:
0.002011096731365496125544890438839229476856739422792843957583386184076397870114370219093085346853618629
comment: $\delta(A_{10}^{*})=\frac{3125\sqrt{11}}{5153632}$; $\det=\frac{1}{11}$, $\mu=\frac{10}{11}$, kissing number $22$
A*
10
density:
0.005128626565043158139175978010541717157430517991691223619855702950705114855293450062702360012617525867
comment: $\Delta(A_{10}^{*})=\frac{625\sqrt{11}\,\pi^{5}}{123687168}$
A*
10
hermite:
1.155437832009218762368668306844038567203526505795372345829318449079678710773745742250058319348359151
comment: $\gamma(A_{10}^{*})=\frac{10}{11^{9/10}}$
A*
11
centre:
0.001048150355161516756873678341173930935669615508805010026688237873741051725255764306928505011101338770
comment: $\delta(A_{11}^{*})=\frac{161051\sqrt{11}}{509607936}$; $\det=\frac{1}{12}$, $\mu=\frac{11}{12}$, kissing number $24$
A*
11
density:
0.001974824150343715469595965443958393041762749590345920360474788772257660917585895503098313902280068258
comment: $\Delta(A_{11}^{*})=\frac{14641\sqrt{11}\,\pi^{5}}{7524679680}$
A*
11
hermite:
1.148996815746165846296960885264741790700002400609945803764768929329747210685100065506658830954360798
comment: $\gamma(A_{11}^{*})=\frac{11}{2^{20/11}\cdot 3^{10/11}}$
A*
12
centre:
0.0005445516654612287734368424986333603722080561303198830716247359856148450674774171952016678743226590763
comment: $\delta(A_{12}^{*})=\frac{729\sqrt{13}}{4826809}$; $\det=\frac{1}{13}$, $\mu=\frac{12}{13}$, kissing number $26$
A*
12
density:
0.0007271195646081384619650923047635419083142301307217549548634271737744865157382171263821803032588546585
comment: $\Delta(A_{12}^{*})=\frac{81\sqrt{13}\,\pi^{6}}{386144720}$
A*
12
hermite:
1.143053364832193642080725451708573220727260945388832966869381431266957926248229779523958599679964714
comment: $\gamma(A_{12}^{*})=\frac{12}{13^{11/12}}$
A*
13
centre:
0.0002821458276187352533077005154439994752258310794600173775156566999963893019384403148510638245706853397
comment: $\delta(A_{13}^{*})=\frac{4826809\sqrt{13}}{61681958912}$; $\det=\frac{1}{14}$, $\mu=\frac{13}{14}$, kissing number $28$
A*
13
density:
0.0002569301036717477422742118134064993713459522224358610766287012328264141886487130659844282293808936008
comment: $\Delta(A_{13}^{*})=\frac{371293\sqrt{13}\,\pi^{6}}{5009249710080}$
A*
13
hermite:
1.137572308593258625637631643430085515600377313572473565145920520158833345042112686021445702022483204
comment: $\gamma(A_{13}^{*})=\frac{13}{2^{12/13}\cdot 7^{12/13}}$
A*
14
centre:
0.0001458421730171785424723908443133514733746765816159557651623415757689252951981278246091254908184875430
comment: $\delta(A_{14}^{*})=\frac{823543\sqrt{15}}{21870000000}$; $\det=\frac{1}{15}$, $\mu=\frac{14}{15}$, kissing number $30$
A*
14
density:
0.00008739804116826102174362552389301032529456474747962392721534278745842314079160982731392819652865701006
comment: $\Delta(A_{14}^{*})=\frac{117649\sqrt{15}\,\pi^{7}}{15746400000000}$
A*
14
hermite:
1.132513258662339501526221675077404393975218377385689528415972103227012265105474198628376870811977194
comment: $\gamma(A_{14}^{*})=\frac{14}{3^{13/14}\cdot 5^{13/14}}$
A*
15
centre:
0.00007523061798547222103549939324306546194602063132948107993189367294286806783946073858887865431546088309
comment: $\delta(A_{15}^{*})=\frac{170859375\sqrt{15}}{8796093022208}$; $\det=\frac{1}{16}$, $\mu=\frac{15}{16}$, kissing number $32$
A*
15
density:
0.00002869621374274322137325433709257593165961052901982261502872546854856614255296226815568877765798637408
comment: $\Delta(A_{15}^{*})=\frac{84375\sqrt{15}\,\pi^{7}}{34394098106368}$
A*
15
hermite:
1.127835971326984360272305102347244485327075963303601726962421015434996163709900834067412006867501465
comment: $\gamma(A_{15}^{*})=\frac{15}{2^{56/15}}$
A*
16
centre:
0.00003873584375114728158351054057544847364196745562186837350337692882035961376851181840819029593599784200
comment: $\delta(A_{16}^{*})=\frac{65536\sqrt{17}}{6975757441}$; $\det=\frac{1}{17}$, $\mu=\frac{16}{17}$, kissing number $34$
A*
16
density:
0.000009115730527441084092317713629740388914018878883339380598266389404523725588153261829420256293581249996
comment: $\Delta(A_{16}^{*})=\frac{512\sqrt{17}\,\pi^{8}}{2197363593915}$
A*
16
hermite:
1.123502695890729540774873028726349966698516820755169676831194169154014775336602344148221496644896615
comment: $\gamma(A_{16}^{*})=\frac{16}{17^{15/16}}$
A*
17
centre:
0.00001991250347508113859352333466441257379435276340674603892331878750254765006236882888081508594596403967
comment: $\delta(A_{17}^{*})=\frac{6975757441\sqrt{17}}{1444408272617472}$; $\det=\frac{1}{18}$, $\mu=\frac{17}{18}$, kissing number $36$
A*
17
density:
0.000002807286781408316638295640060373858120273725992651788442016232554446651885175297198895982283445718239
comment: $\Delta(A_{17}^{*})=\frac{410338673\sqrt{17}\,\pi^{8}}{5718460310160998400}$
A*
17
hermite:
1.119479062427896791535303203006780044804589499279177499125553892945337408812166626662943414206350314
comment: $\gamma(A_{17}^{*})=\frac{17}{2^{16/17}\cdot 3^{32/17}}$
A*
18
centre:
0.00001022132072661282224937805245426146495562922065356840654269212382908442982738599300089075203223598147
comment: $\delta(A_{18}^{*})=\frac{387420489\sqrt{19}}{165216101262848}$; $\det=\frac{1}{19}$, $\mu=\frac{18}{19}$, kissing number $38$
A*
18
density:
8.396394534243116942750547463458954113069587476047609566612383003759869251177299670009908576426216291e-7
comment: $\Delta(A_{18}^{*})=\frac{4782969\sqrt{19}\,\pi^{9}}{740168133657559040}$
A*
18
hermite:
1.115734265234866944854986098392080008053192398765907077007752270042189707449356098498242413059114224
comment: $\gamma(A_{18}^{*})=\frac{18}{19^{17/18}}$
A*
19
centre:
0.000005239855740004983735897934480736998892496144036132633328886814777991680265976657679146434666937127839
comment: $\delta(A_{19}^{*})=\frac{322687697779\sqrt{19}}{268435456000000000}$; $\det=\frac{1}{20}$, $\mu=\frac{19}{20}$, kissing number $40$
A*
19
density:
2.442904637657317290592268067367763165544908550596852830245785646369921430225902433581091123304839847e-7
comment: $\Delta(A_{19}^{*})=\frac{16983563041\sqrt{19}\,\pi^{9}}{9033331507200000000000}$
A*
19
hermite:
1.112240918036640366594758897653426845182016995816149071438909133396902229961636782708622281822396044
comment: $\gamma(A_{19}^{*})=\frac{19}{2^{36/19}\cdot 5^{18/19}}$
A*
20
centre:
0.000002682975725638529010340702806732677732185527920817423622944184459279912668206851494848007254405465140
comment: $\delta(A_{20}^{*})=\frac{9765625\sqrt{21}}{16679880978201}$; $\det=\frac{1}{21}$, $\mu=\frac{20}{21}$, kissing number $42$
A*
20
density:
6.923926315359767924173009961248276615178634727633231369760281101421982357598236884605994803319581915e-8
comment: $\Delta(A_{20}^{*})=\frac{390625\sqrt{21}\,\pi^{10}}{2421118083747831552}$
A*
20
hermite:
1.108974769814499329717170067655231745318239781511967589118407581131761818718317214575653622775611009
comment: $\gamma(A_{20}^{*})=\frac{20}{3^{19/20}\cdot 7^{19/20}}$
A*
21
centre:
0.000001372289933427013491866991487965588803340410888367382136732905824436814799589710791801905117813354906
comment: $\delta(A_{21}^{*})=\frac{16679880978201\sqrt{21}}{55700195201880424448}$; $\det=\frac{1}{22}$, $\mu=\frac{21}{22}$, kissing number $44$
A*
21
density:
1.914227868615882773291375972943886863704713257552811289520862749058894629437533674465434555266016576e-8
comment: $\Delta(A_{21}^{*})=\frac{1400846643\sqrt{21}\,\pi^{10}}{31405472609614928281600}$
A*
21
hermite:
1.105914375737187161985183787204636436466604835115040378925990157963012976242477446872005267850159509
comment: $\gamma(A_{21}^{*})=\frac{21}{2^{20/21}\cdot 11^{20/21}}$
A*
22
centre:
7.012087168578560871576672144298089894766063803032390222252121468217926366300087090203903323415541361e-7
comment: $\delta(A_{22}^{*})=\frac{285311670611\sqrt{23}}{1951354384207722496}$; $\det=\frac{1}{23}$, $\mu=\frac{22}{23}$, kissing number $46$
A*
22
density:
5.168210426133794663051746162163299634697400905956595971034585545256761555050117010062911555681944705e-9
comment: $\Delta(A_{22}^{*})=\frac{25937424601\sqrt{23}\,\pi^{11}}{7081074789412983393484800}$
A*
22
hermite:
1.103040769308036072443570895291827614075395723503595137190776423705099269320424594870338528499041089
comment: $\gamma(A_{22}^{*})=\frac{22}{23^{21/22}}$
A*
23
centre:
3.579781794998931904465208952879284491084851802739349122313187591048840675574681701060438665171695652e-7
comment: $\delta(A_{23}^{*})=\frac{952809757913927\sqrt{23}}{12764786611036820078592}$; $\det=\frac{1}{24}$, $\mu=\frac{23}{24}$, kissing number $48$
A*
23
density:
1.364131836064964407869667097883241434002487344746052085782122991542884838368745210742899466835728432e-9
comment: $\Delta(A_{23}^{*})=\frac{41426511213649\sqrt{23}\,\pi^{11}}{42848392465514395024647782400}$
A*
23
hermite:
1.100337156857078421258574319957845798641363442793016335968019213688914877163919775293714133938567575
comment: $\gamma(A_{23}^{*})=\frac{23}{2^{66/23}\cdot 3^{22/23}}$
A*
24
centre:
2176782336/11920928955078125
comment: $\delta(A_{24}^{*})=\frac{2176782336}{11920928955078125}$; $\det=\frac{1}{25}$, $\mu=\frac{24}{25}$, kissing number $50$
A*
24
density:
3.523436209156009968348354384146930874597117849821738316520513757221591158598065996616521976666113545e-10
comment: $\Delta(A_{24}^{*})=\frac{8748\pi^{12}}{22947788238525390625}$
A*
24
hermite:
1.097788642639603496857716303604875741959840539496487698611056199363117020686748520427747245299641720
comment: $\gamma(A_{24}^{*})=\frac{24}{5^{23/12}}$
A*
3
centre:
0.1623797632095822462681980945161755344008879925447231838802319043236187203352000034763574550084920540
comment: $\delta(A_{3}^{*})=\frac{3\sqrt{3}}{32}$; $\det=\frac{1}{4}$, $\mu=\frac{3}{4}$, kissing number $8$; $A_3^{*}$ is the body-centred cubic lattice
A*
3
density:
0.6801747615878316939727793466158083996065248430347777158387068507705461929223827063815468705795225633
comment: $\Delta(A_{3}^{*})=\frac{\sqrt{3}\,\pi}{8}$; $A_3^{*}$ is the body-centred cubic lattice
A*
3
hermite:
1.190550788976149606063779229454231195293619995924889757356214321368912379218164379955158159946665718
comment: $\gamma(A_{3}^{*})=\frac{3}{2^{4/3}}$; $A_3^{*}$ is the body-centred cubic lattice
A*
4
centre:
0.08944271909999158785636694674925104941762473438446102897083588981642083702551219597657657633515129100
comment: $\delta(A_{4}^{*})=\frac{\sqrt{5}}{25}$; $\det=\frac{1}{5}$, $\mu=\frac{4}{5}$, kissing number $10$
A*
4
density:
0.4413821270373381062068769848516025172876540449080917899638233763535485027739893798021091040066085028
comment: $\Delta(A_{4}^{*})=\frac{\sqrt{5}\,\pi^{2}}{50}$
A*
4
hermite:
1.196279024976976433529519195312730716290767806091765076413274800038337173578397610278899540565990780
comment: $\gamma(A_{4}^{*})=\frac{4}{5^{3/4}}$
A*
5
centre:
0.04852578076171418611999074801934193219272175259573623533573995758269359647651486326854197934849787923
comment: $\delta(A_{5}^{*})=\frac{25\sqrt{5}}{1152}$; $\det=\frac{1}{6}$, $\mu=\frac{5}{6}$, kissing number $12$
A*
5
density:
0.2554294716651262188697204773446773826896146093218123784512866761305257539201327429410353611149354761
comment: $\Delta(A_{5}^{*})=\frac{5\sqrt{5}\,\pi^{2}}{432}$
A*
5
hermite:
1.192474234254379584204353677619264087477208283051068116656973816465235418986027916546740133723444200
comment: $\gamma(A_{5}^{*})=\frac{5}{2^{4/5}\cdot 3^{4/5}}$
A*
6
centre:
0.02603326727359473248671414917939505520924817709301244711003827638426707661341751383000386586545651512
comment: $\delta(A_{6}^{*})=\frac{27\sqrt{7}}{2744}$; $\det=\frac{1}{7}$, $\mu=\frac{6}{7}$, kissing number $14$
A*
6
density:
0.1345324479962121387393057194247756148264690094617972064845010730943881002910914394207990711057535453
comment: $\Delta(A_{6}^{*})=\frac{9\sqrt{7}\,\pi^{3}}{5488}$
A*
6
hermite:
1.185503617944418707977707297315562220751750916751643841759215107891221793818492532732982695834295264
comment: $\gamma(A_{6}^{*})=\frac{6}{7^{5/6}}$
A*
7
centre:
0.01384723968040702167575155950162149545316496123958252581583158446511789865673808722414878479096631559
comment: $\delta(A_{7}^{*})=\frac{343\sqrt{7}}{65536}$; $\det=\frac{1}{8}$, $\mu=\frac{7}{8}$, kissing number $16$
A*
7
density:
0.06542496682500976318841178317049200689781437042894750142896439628175291671274229810027112234011573601
comment: $\Delta(A_{7}^{*})=\frac{49\sqrt{7}\,\pi^{3}}{61440}$
A*
7
hermite:
1.177662668553311615449978528282074166320466148430292132843997740402437682256855158817239670964254833
comment: $\gamma(A_{7}^{*})=\frac{7}{2^{18/7}}$
A*
8
centre:
16/2187
comment: $\delta(A_{8}^{*})=\frac{16}{2187}$; $\det=\frac{1}{9}$, $\mu=\frac{8}{9}$, kissing number $18$
A*
8
density:
0.02969336718000379126244180237424328951371058853000622517648768766650536669923096972721288438797530637
comment: $\Delta(A_{8}^{*})=\frac{2\pi^{4}}{6561}$
A*
8
hermite:
1.169843567068882187394861246041776937920060969071841934872820140974103817953157451600660577996979495
comment: $\gamma(A_{8}^{*})=\frac{8}{3^{7/4}}$
A*
9
centre:
19683/5120000
comment: $\delta(A_{9}^{*})=\frac{19683}{5120000}$; $\det=\frac{1}{10}$, $\mu=\frac{9}{10}$, kissing number $20$
A*
9
density:
0.01268057631496210299024375045179750466090203749203351293090715526701584886214768321832122164538123778
comment: $\Delta(A_{9}^{*})=\frac{729\pi^{4}}{5600000}$
A*
9
hermite:
1.162394698513395487869067991824878788678359736844704259274948851271907040790019706590131801511516636
comment: $\gamma(A_{9}^{*})=\frac{9}{2^{8/9}\cdot 5^{8/9}}$
D
10
centre:
1/64
comment: $\delta(D_{10})=\frac{1}{64}$; $\det=4$, $\mu=2$, kissing number $180$
D
10
density:
0.03984631312308352256025277474523901126045582117553059772831932860497851460073099277158639329082097902
comment: $\Delta(D_{10})=\frac{\pi^{5}}{7680}$
D
10
hermite:
1.741101126592248278272540034959492197958250848696006096483719137013500035504956020376757529743340835
comment: $\gamma(D_{10})=2^{4/5}$
D
11
centre:
0.01104854345603980506876319315788826623882556152638240682169281045305259748798521124101865261193469979
comment: $\delta(D_{11})=\frac{\sqrt{2}}{128}$; $\det=4$, $\mu=2$, kissing number $220$
D
11
density:
0.02081660358713249246851755142559941854696573947086251038321947354173607770740169453703408263008532520
comment: $\Delta(D_{11})=\frac{\sqrt{2}\,\pi^{5}}{20790}$
D
11
hermite:
1.763182509992042383391871761763604832617070787458418968156890322075515373842549813095564618098758414
comment: $\gamma(D_{11})=2^{9/11}$
D
12
centre:
1/128
comment: $\delta(D_{12})=\frac{1}{128}$; $\det=4$, $\mu=2$, kissing number $264$
D
12
density:
0.01043174038167648043652581861602018119430574574740718984311063571488817179733452922633703785972130790
comment: $\Delta(D_{12})=\frac{\pi^{6}}{92160}$
D
12
hermite:
1.781797436280678609480452411181025015974425231756320806767513984503861606631524985275051534501114395
comment: $\gamma(D_{12})=2^{5/6}$
D
13
centre:
0.005524271728019902534381596578944133119412780763191203410846405226526298743992605620509326305967349893
comment: $\delta(D_{13})=\frac{\sqrt{2}}{256}$; $\det=4$, $\mu=2$, kissing number $312$
D
13
density:
0.005030560684771259670778501933859200113077304220606558325382183414811115955907235458098968860901921006
comment: $\Delta(D_{13})=\frac{\sqrt{2}\,\pi^{6}}{270270}$
D
13
hermite:
1.797701946391083748217080870549112685362805768801854913219969080072771197585775002464001567572728231
comment: $\gamma(D_{13})=2^{11/13}$
D
14
centre:
1/256
comment: $\delta(D_{14})=\frac{1}{256}$; $\det=4$, $\mu=2$, kissing number $364$
D
14
density:
0.002340877067659344050342731966976718929747514009250528787476930255133257895657871311491236222012966375
comment: $\Delta(D_{14})=\frac{\pi^{7}}{1290240}$
D
14
hermite:
1.811447328527813343188345746430206375400891762515874710237416262768844934627125673909528787782071557
comment: $\gamma(D_{14})=2^{6/7}$
D
15
centre:
0.002762135864009951267190798289472066559706390381595601705423202613263149371996302810254663152983674947
comment: $\delta(D_{15})=\frac{\sqrt{2}}{512}$; $\det=4$, $\mu=2$, kissing number $420$
D
15
density:
0.001053598166047668602988468974088070001589559639777986597220419155604373309441288488931896571228168658
comment: $\Delta(D_{15})=\frac{\sqrt{2}\,\pi^{7}}{4054050}$
D
15
hermite:
1.823444977116433615632210157088315446532424756712371891434363603649620034805593810642955674651026071
comment: $\gamma(D_{15})=2^{13/15}$
D
16
centre:
1/512
comment: $\delta(D_{16})=\frac{1}{512}$; $\det=4$, $\mu=2$, kissing number $480$
D
16
density:
0.0004596301374197132901208581108895809014552127642745285127238968823177717373236460026833895402705560171
comment: $\Delta(D_{16})=\frac{\pi^{8}}{20643840}$
D
16
hermite:
1.834008086409342463487083189588288856077310332873679499583324138706477662244694725682930978761320627
comment: $\gamma(D_{16})=2^{7/8}$
D
17
centre:
0.001381067932004975633595399144736033279853195190797800852711601306631574685998151405127331576491837473
comment: $\delta(D_{17})=\frac{\sqrt{2}}{1024}$; $\det=4$, $\mu=2$, kissing number $544$
D
17
density:
0.0001947044857818255767834649600340239148692965034703838084254511881296855370101989179443183260286988034
comment: $\Delta(D_{17})=\frac{\sqrt{2}\,\pi^{8}}{68918850}$
D
17
hermite:
1.843379281881730798286822083933046561318432049217838373392893084752710250315544089737509611615431545
comment: $\gamma(D_{17})=2^{15/17}$
D
18
centre:
1/1024
comment: $\delta(D_{18})=\frac{1}{1024}$; $\det=4$, $\mu=2$, kissing number $612$
D
18
density:
0.00008022059239367991093633043508173666485140377031030578429387587028343190143434003721509597996456690973
comment: $\Delta(D_{18})=\frac{\pi^{9}}{371589120}$
D
18
hermite:
1.851749424574580858409381921539531664895374719178026270189414539251873052022230063371359777734573426
comment: $\gamma(D_{18})=2^{8/9}$
D
19
centre:
0.0006905339660024878167976995723680166399265975953989004263558006533157873429990757025636657882459187367
comment: $\delta(D_{19})=\frac{\sqrt{2}}{2048}$; $\det=4$, $\mu=2$, kissing number $684$
D
19
density:
0.00003219379906069271485067828951841922287994012647824365295830751909125131408072314178950984246778347148
comment: $\Delta(D_{19})=\frac{\sqrt{2}\,\pi^{9}}{1309458150}$
D
19
hermite:
1.859270710016812562123254411861692017391741056754022418807623253650390951186565003295635179484872202
comment: $\gamma(D_{19})=2^{17/19}$
D
20
centre:
1/2048
comment: $\delta(D_{20})=\frac{1}{2048}$; $\det=4$, $\mu=2$, kissing number $760$
D
20
density:
0.00001260102118653030274052651086567326643417306691804037401943475220603852018356824016384215735021313726
comment: $\Delta(D_{20})=\frac{\pi^{10}}{7431782400}$
D
20
hermite:
1.866065983073614831962686532299884334054459928702988077800994770970794040621127095207414946789290627
comment: $\gamma(D_{20})=2^{9/10}$
D
21
centre:
0.0003452669830012439083988497861840083199632987976994502131779003266578936714995378512818328941229593683
comment: $\delta(D_{21})=\frac{\sqrt{2}}{4096}$; $\det=4$, $\mu=2$, kissing number $840$
D
21
density:
0.000004816181077153248527235069969072587297983772242769206991389298988054133465637759707629929029225514467
comment: $\Delta(D_{21})=\frac{\sqrt{2}\,\pi^{10}}{27498621150}$
D
21
hermite:
1.872235484907209776378711215229745504726876897943956214601289522670065829941920477891890268718690680
comment: $\gamma(D_{21})=2^{19/21}$
D
22
centre:
1/4096
comment: $\delta(D_{22})=\frac{1}{4096}$; $\det=4$, $\mu=2$, kissing number $924$
D
22
density:
0.000001799421617606042670229270009104194702444247616216063830541953787553098187456566884103255944885519082
comment: $\Delta(D_{22})=\frac{\pi^{11}}{163499212800}$
D
22
hermite:
1.877861821323412700576744871323752854765976936737945629652613867420387780642365977698176175232641815
comment: $\gamma(D_{22})=2^{10/11}$
D
23
centre:
0.0001726334915006219541994248930920041599816493988497251065889501633289468357497689256409164470614796842
comment: $\delta(D_{23})=\frac{\sqrt{2}}{8192}$; $\det=4$, $\mu=2$, kissing number $1012$
D
23
density:
6.578469169714270766418035999085643475314254400414369966793037041392389567346920524014085263231902891e-7
comment: $\Delta(D_{23})=\frac{\sqrt{2}\,\pi^{11}}{632468286450}$
D
23
hermite:
1.883013676297781883635519373298952450378415489447049402291517808983205147730849390317787227894124354
comment: $\gamma(D_{23})=2^{21/23}$
D
24
centre:
1/8192
comment: $\delta(D_{24})=\frac{1}{8192}$; $\det=4$, $\mu=2$, kissing number $1104$
D
24
density:
2.355437389409085751835138627546335009971279071045292178020810079750360881654035072616391233137382707e-7
comment: $\Delta(D_{24})=\frac{\pi^{12}}{3923981107200}$
D
24
hermite:
1.887748625363386993283826313335068752015136606677485627484250284636572975477413406090396909221235416
comment: $\gamma(D_{24})=2^{11/12}$
D
4
centre:
1/8
comment: $\delta(D_{4})=\frac{1}{8}$; $\det=4$, $\mu=2$, kissing number $24$; $D_{4}=\Lambda_{4}$
D
4
density:
0.6168502750680849136771556874922594459571062129525494141508343360137528014012003276876108377324095145
comment: $\Delta(D_{4})=\frac{\pi^{2}}{16}$; the densest lattice packing in dimension 4 [8]; $D_{4}=\Lambda_{4}$
D
4
hermite:
1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
comment: $\gamma(D_{4})=\sqrt{2}$; Hermite's constant $\gamma_{4}$ [8]; $D_{4}=\Lambda_{4}$
equals: Algebraic_numbers_of_degree_2#1,0,-2,2
D
5
centre:
0.08838834764831844055010554526310612991060449221105925457354248362442077990388168992814922089547759830
comment: $\delta(D_{5})=\frac{\sqrt{2}}{16}$; $\det=4$, $\mu=2$, kissing number $40$; $D_{5}=\Lambda_{5}$
D
5
density:
0.4652576133092586356105040624112936859946577513965361577435664445013271841888718143111600891540540958
comment: $\Delta(D_{5})=\frac{\sqrt{2}\,\pi^{2}}{30}$; the densest lattice packing in dimension 5 [8]; $D_{5}=\Lambda_{5}$
D
5
hermite:
1.515716566510398082347259801306445238681283542978141642037505242097453677202058277641176134849431791
comment: $\gamma(D_{5})=2^{3/5}$; Hermite's constant $\gamma_{5}$ [8]; $D_{5}=\Lambda_{5}$
D
6
centre:
1/16
comment: $\delta(D_{6})=\frac{1}{16}$; $\det=4$, $\mu=2$, kissing number $60$
D
6
density:
0.3229820487531231268278782819489728666898467558946365384806722719146499470569344378913198024229467312
comment: $\Delta(D_{6})=\frac{\pi^{3}}{96}$
D
6
hermite:
1.587401051968199474751705639272308260391493327899853009808285761825216505624219173273544213262220957
comment: $\gamma(D_{6})=2^{2/3}$
D
7
centre:
0.04419417382415922027505277263155306495530224610552962728677124181221038995194084496407461044773879915
comment: $\delta(D_{7})=\frac{\sqrt{2}}{32}$; $\det=4$, $\mu=2$, kissing number $84$
D
7
density:
0.2088071285712982487119759477845852987347505409091042469244108708947825315491728189023710095441813824
comment: $\Delta(D_{7})=\frac{\sqrt{2}\,\pi^{3}}{210}$
D
7
hermite:
1.640670712015275862340569365734992099789404915306025246598383626990030803453104404313461505028108899
comment: $\gamma(D_{7})=2^{5/7}$
D
8
centre:
1/32
comment: $\delta(D_{8})=\frac{1}{32}$; $\det=4$, $\mu=2$, kissing number $112$
D
8
density:
0.1268347539505240068182816831884181136064161271779758094940987752473578847094104117058878479619179591
comment: $\Delta(D_{8})=\frac{\pi^{4}}{768}$
D
8
hermite:
1.681792830507429086062250952466429790080068524713569021626452171949849509907804479628648008398585072
comment: $\gamma(D_{8})=2^{3/4}$
D
9
centre:
0.02209708691207961013752638631577653247765112305276481364338562090610519497597042248203730522386939957
comment: $\delta(D_{9})=\frac{\sqrt{2}}{64}$; $\det=4$, $\mu=2$, kissing number $144$
D
9
density:
0.07288743790408555526660687722882599977128699625536283354816057423198158865453764503620500157335241281
comment: $\Delta(D_{9})=\frac{\sqrt{2}\,\pi^{4}}{1890}$
D
9
hermite:
1.714487965706145661766070110832072051511440720277524415717680108293178339505411178988245367996736297
comment: $\gamma(D_{9})=2^{7/9}$
D*
10
centre:
1/512
comment: $\delta(D_{10}^{*})=\frac{1}{512}$; $\det=\frac{1}{4}$, $\mu=1$, kissing number $20$
D*
10
density:
0.004980789140385440320031596843154876407556977646941324716039916075622314325091374096448299161352622378
comment: $\Delta(D_{10}^{*})=\frac{\pi^{5}}{61440}$
D*
10
hermite:
1.148698354997035006798626946777927589443850889097797505513711118493603206253513056811473113011508474
comment: $\gamma(D_{10}^{*})=2^{1/5}$
D*
11
centre:
1/1024
comment: $\delta(D_{11}^{*})=\frac{1}{1024}$; $\det=\frac{1}{4}$, $\mu=1$, kissing number $22$
D*
11
density:
0.001839945194716699454441686568322724906687714686315987196690070007733495623698978021198794495391444919
comment: $\Delta(D_{11}^{*})=\frac{\pi^{5}}{166320}$
D*
11
hermite:
1.134312522195462580992497732917119138938301920796695962016192664318287105778049009048037048416301812
comment: $\gamma(D_{11}^{*})=2^{2/11}$
D*
12
centre:
1/2048
comment: $\delta(D_{12}^{*})=\frac{1}{2048}$; $\det=\frac{1}{4}$, $\mu=1$, kissing number $24$
D*
12
density:
0.0006519837738547800272828636635012613246441091092129493651944147321805107373334080766460648662325817438
comment: $\Delta(D_{12}^{*})=\frac{\pi^{6}}{1474560}$
D*
12
hermite:
1.122462048309372981433533049679179516232411110613986753440409545882904005565861247087923227112509081
comment: $\gamma(D_{12}^{*})=2^{1/6}$
D*
13
centre:
1/4096
comment: $\delta(D_{13}^{*})=\frac{1}{4096}$; $\det=\frac{1}{4}$, $\mu=1$, kissing number $26$
D*
13
density:
0.0002223214733357624868257250720696941713072153605841026140389879073436040609488544357261306769970575110
comment: $\Delta(D_{13}^{*})=\frac{\pi^{6}}{4324320}$
D*
13
hermite:
1.112531476096486920262088977149391353845053178180482097211804844090556252100582416200587768880471251
comment: $\gamma(D_{13}^{*})=2^{2/13}$
D*
14
centre:
1/8192
comment: $\delta(D_{14}^{*})=\frac{1}{8192}$; $\det=\frac{1}{4}$, $\mu=1$, kissing number $28$
D*
14
density:
0.00007315240836435450157321037396802246655460981278907902460865407047291430923930847848410113193790519922
comment: $\Delta(D_{14}^{*})=\frac{\pi^{7}}{41287680}$
D*
14
hermite:
1.104089513673812337649505387623344721325326600780124165514532464142106322880380980716598289886302005
comment: $\gamma(D_{14}^{*})=2^{1/7}$
D*
15
centre:
1/16384
comment: $\delta(D_{15}^{*})=\frac{1}{16384}$; $\det=\frac{1}{4}$, $\mu=1$, kissing number $30$
D*
15
density:
0.00002328145024556301775010642515718881297650985184025389936262992017537350510056002547559271456236672075
comment: $\Delta(D_{15}^{*})=\frac{\pi^{7}}{129729600}$
D*
15
hermite:
1.096824979694625960681471778028803387827847321527142350042026590511525718728112099321909227103947672
comment: $\gamma(D_{15}^{*})=2^{2/15}$
D*
16
centre:
1/32768
comment: $\delta(D_{16}^{*})=\frac{1}{32768}$; $\det=\frac{1}{4}$, $\mu=1$, kissing number $32$
D*
16
density:
0.000007181720897183020158138407982649701585237699441789508011310888786215183395681968791927961566727437766
comment: $\Delta(D_{16}^{*})=\frac{\pi^{8}}{1321205760}$
D*
16
hermite:
1.090507732665257659207010655760707978992702718540067121785667647683300530848841840338211140494203120
comment: $\gamma(D_{16}^{*})=2^{1/8}$
D*
17
centre:
1/65536
comment: $\delta(D_{17}^{*})=\frac{1}{65536}$; $\det=\frac{1}{4}$, $\mu=1$, kissing number $34$
D*
17
density:
0.000002151200972246384245549425044796064002423044337570808524289366093027095658025565641701133000763514610
comment: $\Delta(D_{17}^{*})=\frac{\pi^{8}}{4410806400}$
D*
17
hermite:
1.084963913643637129667077523875454847611686369477650775238069264432332883787604319563593590399253638
comment: $\gamma(D_{17}^{*})=2^{2/17}$
D*
18
centre:
1/131072
comment: $\delta(D_{18}^{*})=\frac{1}{131072}$; $\det=\frac{1}{4}$, $\mu=1$, kissing number $36$
D*
18
density:
6.267233780756243041900815240760676941515919555492639397959052365893117299557815407429373434731789822e-7
comment: $\Delta(D_{18}^{*})=\frac{\pi^{9}}{47563407360}$
D*
18
hermite:
1.080059738892306169872930831288596912737466762465644733136497122283534876578258927885325469533216098
comment: $\gamma(D_{18}^{*})=2^{1/9}$
D*
19
centre:
1/262144
comment: $\delta(D_{19}^{*})=\frac{1}{262144}$; $\det=\frac{1}{4}$, $\mu=1$, kissing number $38$
D*
19
density:
1.778472939685384589378474648156451640170548416240947479203466002083488862472974043782248662749063967e-7
comment: $\Delta(D_{19}^{*})=\frac{\pi^{9}}{167610643200}$
D*
19
hermite:
1.075690586220182474231400776464092135133759903355640199420412371622904977065515069260940075713230671
comment: $\gamma(D_{19}^{*})=2^{2/19}$
D*
20
centre:
1/524288
comment: $\delta(D_{20}^{*})=\frac{1}{524288}$; $\det=\frac{1}{4}$, $\mu=1$, kissing number $40$
D*
20
density:
4.922273900988399508018168306903619700848854264859521101341700080483796946706343814000842714927006742e-8
comment: $\Delta(D_{20}^{*})=\frac{\pi^{10}}{1902536294400}$
D*
20
hermite:
1.071773462536293164213006325023342022906384604977556783482780668144245438837468954516907443983687770
comment: $\gamma(D_{20}^{*})=2^{1/10}$
D*
21
centre:
1/1048576
comment: $\delta(D_{21}^{*})=\frac{1}{1048576}$; $\det=\frac{1}{4}$, $\mu=1$, kissing number $42$
D*
21
density:
1.330294648077106588591918620744521198986072804808539550841437845846974972187139787743151495214615370e-8
comment: $\Delta(D_{21}^{*})=\frac{\pi^{10}}{7039647014400}$
D*
21
hermite:
1.068241690814402220013606897069232472303291177670581722176337650608827204480478972033118520513530295
comment: $\gamma(D_{21}^{*})=2^{2/21}$
D*
22
centre:
1/2097152
comment: $\delta(D_{22}^{*})=\frac{1}{2097152}$; $\det=\frac{1}{4}$, $\mu=1$, kissing number $44$
D*
22
density:
3.514495346886802090291542986531630278211421125421999669027253491314644897376107195514171767354529458e-9
comment: $\Delta(D_{22}^{*})=\frac{\pi^{11}}{83711596953600}$
D*
22
hermite:
1.065041089439962678190592595398204490092328886510584811019914747456601900170422889132828048742624901
comment: $\gamma(D_{22}^{*})=2^{1/11}$
D*
23
centre:
1/4194304
comment: $\delta(D_{23}^{*})=\frac{1}{4194304}$; $\det=\frac{1}{4}$, $\mu=1$, kissing number $46$
D*
23
density:
9.085312811975777010472139868275328175361231916133286997125711576079548388754319841122253240630454458e-10
comment: $\Delta(D_{23}^{*})=\frac{\pi^{11}}{323823762662400}$
D*
23
hermite:
1.062127176862690915800524153411598751007243459063598140151676822461795332663241154793066281672813239
comment: $\gamma(D_{23}^{*})=2^{2/23}$
D*
24
centre:
1/8388608
comment: $\delta(D_{24}^{*})=\frac{1}{8388608}$; $\det=\frac{1}{4}$, $\mu=1$, kissing number $48$
D*
24
density:
2.300231825594810304526502565963217783175077217817668142598447343506211798490268625601944563610725300e-10
comment: $\Delta(D_{24}^{*})=\frac{\pi^{12}}{4018156653772800}$
D*
24
hermite:
1.059463094359295264561825294946341700779204317494185628559208431458761646063255722383768376863945569
comment: $\gamma(D_{24}^{*})=2^{1/12}$
D*
5
centre:
1/16
comment: $\delta(D_{5}^{*})=\frac{1}{16}$; $\det=\frac{1}{4}$, $\mu=1$, kissing number $10$
D*
5
density:
0.3289868133696452872944830333292050378437899802413596875471116458740014940806401747667257801239517411
comment: $\Delta(D_{5}^{*})=\frac{\pi^{2}}{30}$
D*
5
hermite:
1.319507910772894259374001971229640133033469013193418681505807795980535980893520830050353795235959499
comment: $\gamma(D_{5}^{*})=2^{2/5}$
D*
6
centre:
1/32
comment: $\delta(D_{6}^{*})=\frac{1}{32}$; $\det=\frac{1}{4}$, $\mu=1$, kissing number $12$
D*
6
density:
0.1614910243765615634139391409744864333449233779473182692403361359573249735284672189456599012114733656
comment: $\Delta(D_{6}^{*})=\frac{\pi^{3}}{192}$
D*
6
hermite:
1.259921049894873164767210607278228350570251464701507980081975112155299676513959483729396562436255094
comment: $\gamma(D_{6}^{*})=2^{1/3}$
D*
7
centre:
1/64
comment: $\delta(D_{7}^{*})=\frac{1}{64}$; $\det=\frac{1}{4}$, $\mu=1$, kissing number $14$
D*
7
density:
0.07382446828642814327494360730262236952910782991877406593843937643763427361301358580373024055381639570
comment: $\Delta(D_{7}^{*})=\frac{\pi^{3}}{420}$
D*
7
hermite:
1.219013654204475440911691002592560857277411935859960806590971514832067295459667993817258140313705125
comment: $\gamma(D_{7}^{*})=2^{2/7}$
D*
8
centre:
1/128
comment: $\delta(D_{8}^{*})=\frac{1}{128}$; $\det=\frac{1}{4}$, $\mu=1$, kissing number $16$
D*
8
density:
0.03170868848763100170457042079710452840160403179449395237352469381183947117735260292647196199047948976
comment: $\Delta(D_{8}^{*})=\frac{\pi^{4}}{3072}$
D*
8
hermite:
1.189207115002721066717499970560475915292972092463817413019002224719466668226917159870781344538137674
comment: $\gamma(D_{8}^{*})=2^{1/4}$
D*
9
centre:
1/256
comment: $\delta(D_{9}^{*})=\frac{1}{256}$; $\det=\frac{1}{4}$, $\mu=1$, kissing number $18$
D*
9
density:
0.01288480040132307370852385353025199884255655895141341556765447875528715019270201007805844804692499901
comment: $\Delta(D_{9}^{*})=\frac{\pi^{4}}{7560}$
D*
9
hermite:
1.166529039576116580893692634660668061417909153898898354909265761772354220490178139945062540827940208
comment: $\gamma(D_{9}^{*})=2^{2/9}$
E
6
centre:
0.07216878364870322056364359756274468195595021890876585950232529081049720903786666821171442444821869065
comment: $\delta(E_{6})=\frac{\sqrt{3}}{24}$; $\det=3$, $\mu=2$, kissing number $72$; $E_{6}=\Lambda_{6}$
E
6
density:
0.3729475455820649395634775586799581063936647972683873631114040655972831720296832195225267216353405428
comment: $\Delta(E_{6})=\frac{\sqrt{3}\,\pi^{3}}{144}$; the densest lattice packing in dimension 6 [2]; $E_{6}=\Lambda_{6}$
E
6
hermite:
1.665366355311208639217572725017671513324124095787337672980480482451078485985633426184051324797373192
comment: $\gamma(E_{6})=\frac{2}{3^{1/6}}$; Hermite's constant $\gamma_{6}$ [2]; $E_{6}=\Lambda_{6}$
E
7
centre:
1/16
comment: $\delta(E_{7})=\frac{1}{16}$; $\det=2$, $\mu=2$, kissing number $126$; $E_{7}=\Lambda_{7}$
E
7
density:
0.2952978731457125730997744292104894781164313196750962637537575057505370944520543432149209622152655828
comment: $\Delta(E_{7})=\frac{\pi^{3}}{105}$; the densest lattice packing in dimension 7 [2]; $E_{7}=\Lambda_{7}$
E
7
hermite:
1.811447328527813343188345746430206375400891762515874710237416262768844934627125673909528787782071557
comment: $\gamma(E_{7})=2^{6/7}$; Hermite's constant $\gamma_{7}$ [2]; $E_{7}=\Lambda_{7}$
E
8
centre:
1/16
comment: $\delta(E_{8})=\frac{1}{16}$; $\det=1$, $\mu=2$, kissing number $240$; $E_{8}=\Lambda_{8}$
E
8
density:
0.2536695079010480136365633663768362272128322543559516189881975504947157694188208234117756959238359181
comment: $\Delta(E_{8})=\frac{\pi^{4}}{384}$; the densest packing of any kind in dimension 8 [14]; $E_{8}=\Lambda_{8}$
E
8
hermite:
2
comment: $\gamma(E_{8})=2$; Hermite's constant $\gamma_{8}$ [2]; $E_{8}=\Lambda_{8}$
E*
6
centre:
0.06415002990995841827879430894466193951640019458556965289095581405377529692254814952152393284286105836
comment: $\delta(E_{6}^{*})=\frac{\sqrt{3}}{27}$; $\det=\frac{1}{3}$, $\mu=\frac{4}{3}$, kissing number $54$
E*
6
density:
0.3315089294062799462786467188266294279054798197941221005434702805309183751374961951311348636758582602
comment: $\Delta(E_{6}^{*})=\frac{\sqrt{3}\,\pi^{3}}{162}$
E*
6
hermite:
1.601249273568003635567287183132723795064543224301309633357828520598940805358792781392989051159671462
comment: $\gamma(E_{6}^{*})=\frac{4}{3^{5/6}}$
E*
7
centre:
0.04566930840269500676293071408267436905024974790320339546631522309101776509427500097772553422113839018
comment: $\delta(E_{7}^{*})=\frac{27\sqrt{3}}{1024}$; $\det=\frac{1}{2}$, $\mu=\frac{3}{2}$, kissing number $56$
E*
7
density:
0.2157767942296232864617263018076900472706203469909955458001694950955709781028881484380333175175898855
comment: $\Delta(E_{7}^{*})=\frac{9\sqrt{3}\,\pi^{3}}{2240}$
E*
7
hermite:
1.656134270510718506474258081435017081987989901170186248271798696213159484320571471074897434829453008
comment: $\gamma(E_{7}^{*})=\frac{3}{2^{6/7}}$
K
12
centre:
1/27
comment: $\delta(K_{12})=\frac{1}{27}$; $\det=729$, $\mu=4$, kissing number $756$; $K_{12}$ is the Coxeter–Todd lattice
K
12
density:
0.04945417662424405540278906603150308121744946132104149258956153227798837000217850892485706837201212635
comment: $\Delta(K_{12})=\frac{\pi^{6}}{19440}$; the densest lattice packing known in dimension 12 [16]; $K_{12}$ is the Coxeter–Todd lattice
K
12
hermite:
2.309401076758503058036595122007829822590407005080507504074409305935910689211733382774861582342998101
comment: $\gamma(K_{12})=\frac{4}{\sqrt{3}}$; $K_{12}$ is the Coxeter–Todd lattice
Lambda
10
centre:
0.03608439182435161028182179878137234097797510945438292975116264540524860451893333410585721222410934533
comment: $\delta(\Lambda_{10})=\frac{\sqrt{3}}{48}$; $\det=768$, $\mu=4$, kissing number $336$
Lambda
10
density:
0.09202111843130555779386059577030574946623950376510629227349760612731057277995423712397819980248230119
comment: $\Delta(\Lambda_{10})=\frac{\sqrt{3}\,\pi^{5}}{5760}$; the densest lattice packing known in dimension 10 [16]
Lambda
10
hermite:
2.058372017929521168115134100140546258980859052190904222384855679777514164955808108505955673298293323
comment: $\gamma(\Lambda_{10})=\frac{2^{6/5}}{3^{1/10}}$
Lambda
11
centre:
1/32
comment: $\delta(\Lambda_{11})=\frac{1}{32}$; $\det=1024$, $\mu=4$, kissing number $438$
Lambda
11
density:
0.05887824623093438254213397018632719701400686996211159029408224024747185995836729667836142385252623740
comment: $\Delta(\Lambda_{11})=\frac{2\pi^{5}}{10395}$; not the densest lattice packing known in dimension 11: $K_{11}$ has centre density $\frac{\sqrt{3}}{54}$
Lambda
11
hermite:
2.130082178879925356381185190796408980184657773021169622039829494913203800340845778265656097485249802
comment: $\gamma(\Lambda_{11})=2^{12/11}$
Lambda
12
centre:
1/32
comment: $\delta(\Lambda_{12})=\frac{1}{32}$; $\det=1024$, $\mu=4$, kissing number $648$
Lambda
12
density:
0.04172696152670592174610327446408072477722298298962875937244254285955268718933811690534815143888523160
comment: $\Delta(\Lambda_{12})=\frac{\pi^{6}}{23040}$; not the densest lattice packing known in dimension 12: $K_{12}$ has centre density $\frac{1}{27}$
Lambda
12
hermite:
2.244924096618745962867066099358359032464822221227973506880819091765808011131722494175846454225018161
comment: $\gamma(\Lambda_{12})=2^{7/6}$
Lambda
13
centre:
1/32
comment: $\delta(\Lambda_{13})=\frac{1}{32}$; $\det=1024$, $\mu=4$, kissing number $906$
Lambda
13
density:
0.02845714858697759831369280922492085392732356615476513459699045213998131980145336777294472665562336141
comment: $\Delta(\Lambda_{13})=\frac{4\pi^{6}}{135135}$; not the densest lattice packing known in dimension 13: $K_{13}$ has centre density $\frac{\sqrt{3}}{54}$
Lambda
13
hermite:
2.346920920009252782190883505970653978257647218823348190514280236112778699531056704530044027110891233
comment: $\gamma(\Lambda_{13})=2^{16/13}$
Lambda
14
centre:
0.03608439182435161028182179878137234097797510945438292975116264540524860451893333410585721222410934533
comment: $\delta(\Lambda_{14})=\frac{\sqrt{3}}{48}$; $\det=768$, $\mu=4$, kissing number $1422$
Lambda
14
density:
0.02162409608244710546249221367868409564177065936712948464424487710130327456120990167726026005937182741
comment: $\Delta(\Lambda_{14})=\frac{\sqrt{3}\,\pi^{7}}{241920}$; the densest lattice packing known in dimension 14 [16]
Lambda
14
hermite:
2.488643919822375497998830713027331282995624292467588082920749685538170938078947312586119317650029594
comment: $\gamma(\Lambda_{14})=\frac{2^{10/7}}{3^{1/14}}$
Lambda
15
centre:
0.04419417382415922027505277263155306495530224610552962728677124181221038995194084496407461044773879915
comment: $\delta(\Lambda_{15})=\frac{\sqrt{2}}{32}$; $\det=512$, $\mu=4$, kissing number $2340$
Lambda
15
density:
0.01685757065676269764781550358540912002543295423644778555552670648966997295106061582291034513965069853
comment: $\Delta(\Lambda_{15})=\frac{8\sqrt{2}\,\pi^{7}}{2027025}$; the densest lattice packing known in dimension 15 [16]
Lambda
15
hermite:
2.639015821545788518748003942459280266066938026386837363011615591961071961787041660100707590471918997
comment: $\gamma(\Lambda_{15})=2^{7/5}$
Lambda
16
centre:
1/16
comment: $\delta(\Lambda_{16})=\frac{1}{16}$; $\det=256$, $\mu=4$, kissing number $4320$; $\Lambda_{16}=BW_{16}$, the Barnes–Wall lattice
Lambda
16
density:
0.01470816439743082528386745954846658884656680845678491240716470023416869559435667208586846528865779255
comment: $\Delta(\Lambda_{16})=\frac{\pi^{8}}{645120}$; the densest lattice packing known in dimension 16 [16]; $\Lambda_{16}=BW_{16}$, the Barnes–Wall lattice
Lambda
16
hermite:
2.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145
comment: $\gamma(\Lambda_{16})=2^{3/2}$; $\Lambda_{16}=BW_{16}$, the Barnes–Wall lattice
Lambda
17
centre:
1/16
comment: $\delta(\Lambda_{17})=\frac{1}{16}$; $\det=256$, $\mu=4$, kissing number $5346$
Lambda
17
density:
0.008811319182321189869770444983484678153924789606690031715489243517038983815272716868407840771127355841
comment: $\Delta(\Lambda_{17})=\frac{32\pi^{8}}{34459425}$; the densest lattice packing known in dimension 17 [16]
Lambda
17
hermite:
2.886681154059912883177543043257111681405167207157733637364231605191669617520219793568933482805778991
comment: $\gamma(\Lambda_{17})=2^{26/17}$
Lambda
18
centre:
0.07216878364870322056364359756274468195595021890876585950232529081049720903786666821171442444821869065
comment: $\delta(\Lambda_{18})=\frac{\sqrt{3}}{24}$; $\det=192$, $\mu=4$, kissing number $7398$
Lambda
18
density:
0.005928368718469419730049891486523909806100960042824357061950447334418422452973803955425827012554145075
comment: $\Delta(\Lambda_{18})=\frac{\sqrt{3}\,\pi^{9}}{8709120}$; the densest lattice packing known in dimension 18 [16]
Lambda
18
hermite:
2.986825999361043925804195343609699263561550450174920817714066244307066740702547366886684630008909785
comment: $\gamma(\Lambda_{18})=\frac{2^{5/3}}{3^{1/18}}$
Lambda
19
centre:
0.08838834764831844055010554526310612991060449221105925457354248362442077990388168992814922089547759830
comment: $\delta(\Lambda_{19})=\frac{\sqrt{2}}{16}$; $\det=128$, $\mu=4$, kissing number $10668$
Lambda
19
density:
0.004120806279768667500886821058357660528632336189215187578663362443680168202332562149057259835876284349
comment: $\Delta(\Lambda_{19})=\frac{64\sqrt{2}\,\pi^{9}}{654729075}$; the densest lattice packing known in dimension 19 [16]
Lambda
19
hermite:
3.098519284533311451212543035603137070013123293883257942512309292804881953057023682104046634319766642
comment: $\gamma(\Lambda_{19})=2^{31/19}$
Lambda
20
centre:
1/8
comment: $\delta(\Lambda_{20})=\frac{1}{8}$; $\det=64$, $\mu=4$, kissing number $17400$
Lambda
20
density:
0.003225861423751757501574786781612356207148305131018335748975296564745861166993469481943592281654563139
comment: $\Delta(\Lambda_{20})=\frac{\pi^{10}}{29030400}$; the densest lattice packing known in dimension 20 [16]
Lambda
20
hermite:
3.249009585424942090438837531101126605140819897728637770129974852216882973740059542498944519264115310
comment: $\gamma(\Lambda_{20})=2^{17/10}$
Lambda
21
centre:
0.1767766952966368811002110905262122598212089844221185091470849672488415598077633798562984417909551966
comment: $\delta(\Lambda_{21})=\frac{\sqrt{2}}{8}$; $\det=32$, $\mu=4$, kissing number $27720$
Lambda
21
density:
0.002465884711502463245944355824165164696567691388297833979591321081883716334406532970306523662963463407
comment: $\Delta(\Lambda_{21})=\frac{256\sqrt{2}\,\pi^{10}}{13749310575}$; the densest lattice packing known in dimension 21 [16]
Lambda
21
hermite:
3.391455967510140347881421973682253209168920269455297005724336499334965922559574004478846549769583126
comment: $\gamma(\Lambda_{21})=2^{37/21}$
Lambda
22
centre:
0.2886751345948128822545743902509787278238008756350634380093011632419888361514666728468576977928747626
comment: $\delta(\Lambda_{22})=\frac{\sqrt{3}}{6}$; $\det=12$, $\mu=4$, kissing number $49896$
Lambda
22
density:
0.002127660145275864210626933444611565330655697515338490683084571822978614708729444826368239332278713761
comment: $\Delta(\Lambda_{22})=\frac{\sqrt{3}\,\pi^{11}}{239500800}$; the densest lattice packing known in dimension 22 [16]
Lambda
22
hermite:
3.572780195142164825133666976858288632976880296572663439444579656875091486429634301963404042025028554
comment: $\gamma(\Lambda_{22})=\frac{2^{21/11}}{3^{1/22}}$
Lambda
23
centre:
1/2
comment: $\delta(\Lambda_{23})=\frac{1}{2}$; $\det=4$, $\mu=4$, kissing number $93150$
Lambda
23
density:
0.001905328193426062470906566906903334103361515823538275509259618028319837706257289936344921562809463883
comment: $\Delta(\Lambda_{23})=\frac{2048\pi^{11}}{316234143225}$; the densest lattice packing known in dimension 23 [16]
Lambda
23
hermite:
3.766027352595563767271038746597904900756830978894098804583035617966410295461698780635574455788248708
comment: $\gamma(\Lambda_{23})=2^{44/23}$
Lambda
24
centre:
1
comment: $\delta(\Lambda_{24})=1$; $\det=1$, $\mu=4$, kissing number $196560$; $\Lambda_{24}$ is the Leech lattice
equals: One
Lambda
24
density:
0.001929574309403923047903345563685957640168471815000303352234647617331495634250985531487347698186143913
comment: $\Delta(\Lambda_{24})=\frac{\pi^{12}}{479001600}$; the densest packing of any kind in dimension 24 [3]; $\Lambda_{24}$ is the Leech lattice
Lambda
24
hermite:
4
comment: $\gamma(\Lambda_{24})=4$; Hermite's constant $\gamma_{24}$ [4]; $\Lambda_{24}$ is the Leech lattice
Lambda
9
centre:
0.04419417382415922027505277263155306495530224610552962728677124181221038995194084496407461044773879915
comment: $\delta(\Lambda_{9})=\frac{\sqrt{2}}{32}$; $\det=512$, $\mu=4$, kissing number $272$
Lambda
9
density:
0.1457748758081711105332137544576519995425739925107256670963211484639631773090752900724100031467048256
comment: $\Delta(\Lambda_{9})=\frac{\sqrt{2}\,\pi^{4}}{945}$; the densest lattice packing known in dimension 9 [16]
Lambda
9
hermite:
2
comment: $\gamma(\Lambda_{9})=2$
Z
1
centre:
1/2
comment: $\delta(\mathbb{Z})=\frac{1}{2}$; $\det=1$, $\mu=1$, kissing number $2$; $\mathbb{Z}=\Lambda_{1}$
Z
1
density:
1
comment: $\Delta(\mathbb{Z})=1$; the densest packing of any kind in dimension 1; $\mathbb{Z}=\Lambda_{1}$
equals: One
Z
1
hermite:
1
comment: $\gamma(\mathbb{Z})=1$; Hermite's constant $\gamma_{1}$; $\mathbb{Z}=\Lambda_{1}$
equals: One
Z
10
centre:
1/1024
comment: $\delta(\mathbb{Z}^{10})=\frac{1}{1024}$; $\det=1$, $\mu=1$, kissing number $20$
Z
10
density:
0.002490394570192720160015798421577438203778488823470662358019958037811157162545687048224149580676311189
comment: $\Delta(\mathbb{Z}^{10})=\frac{\pi^{5}}{122880}$
Z
10
hermite:
1
comment: $\gamma(\mathbb{Z}^{10})=1$
equals: One
Z
11
centre:
1/2048
comment: $\delta(\mathbb{Z}^{11})=\frac{1}{2048}$; $\det=1$, $\mu=1$, kissing number $22$
Z
11
density:
0.0009199725973583497272208432841613624533438573431579935983450350038667478118494890105993972476957224594
comment: $\Delta(\mathbb{Z}^{11})=\frac{\pi^{5}}{332640}$
Z
11
hermite:
1
comment: $\gamma(\mathbb{Z}^{11})=1$
equals: One
Z
12
centre:
1/4096
comment: $\delta(\mathbb{Z}^{12})=\frac{1}{4096}$; $\det=1$, $\mu=1$, kissing number $24$
Z
12
density:
0.0003259918869273900136414318317506306623220545546064746825972073660902553686667040383230324331162908719
comment: $\Delta(\mathbb{Z}^{12})=\frac{\pi^{6}}{2949120}$
Z
12
hermite:
1
comment: $\gamma(\mathbb{Z}^{12})=1$
equals: One
Z
13
centre:
1/8192
comment: $\delta(\mathbb{Z}^{13})=\frac{1}{8192}$; $\det=1$, $\mu=1$, kissing number $26$
Z
13
density:
0.0001111607366678812434128625360348470856536076802920513070194939536718020304744272178630653384985287555
comment: $\Delta(\mathbb{Z}^{13})=\frac{\pi^{6}}{8648640}$
Z
13
hermite:
1
comment: $\gamma(\mathbb{Z}^{13})=1$
equals: One
Z
14
centre:
1/16384
comment: $\delta(\mathbb{Z}^{14})=\frac{1}{16384}$; $\det=1$, $\mu=1$, kissing number $28$
Z
14
density:
0.00003657620418217725078660518698401123327730490639453951230432703523645715461965423924205056596895259961
comment: $\Delta(\mathbb{Z}^{14})=\frac{\pi^{7}}{82575360}$
Z
14
hermite:
1
comment: $\gamma(\mathbb{Z}^{14})=1$
equals: One
Z
15
centre:
1/32768
comment: $\delta(\mathbb{Z}^{15})=\frac{1}{32768}$; $\det=1$, $\mu=1$, kissing number $30$
Z
15
density:
0.00001164072512278150887505321257859440648825492592012694968131496008768675255028001273779635728118336037
comment: $\Delta(\mathbb{Z}^{15})=\frac{\pi^{7}}{259459200}$
Z
15
hermite:
1
comment: $\gamma(\mathbb{Z}^{15})=1$
equals: One
Z
16
centre:
1/65536
comment: $\delta(\mathbb{Z}^{16})=\frac{1}{65536}$; $\det=1$, $\mu=1$, kissing number $32$
Z
16
density:
0.000003590860448591510079069203991324850792618849720894754005655444393107591697840984395963980783363718883
comment: $\Delta(\mathbb{Z}^{16})=\frac{\pi^{8}}{2642411520}$
Z
16
hermite:
1
comment: $\gamma(\mathbb{Z}^{16})=1$
equals: One
Z
17
centre:
1/131072
comment: $\delta(\mathbb{Z}^{17})=\frac{1}{131072}$; $\det=1$, $\mu=1$, kissing number $34$
Z
17
density:
0.000001075600486123192122774712522398032001211522168785404262144683046513547829012782820850566500381757305
comment: $\Delta(\mathbb{Z}^{17})=\frac{\pi^{8}}{8821612800}$
Z
17
hermite:
1
comment: $\gamma(\mathbb{Z}^{17})=1$
equals: One
Z
18
centre:
1/262144
comment: $\delta(\mathbb{Z}^{18})=\frac{1}{262144}$; $\det=1$, $\mu=1$, kissing number $36$
Z
18
density:
3.133616890378121520950407620380338470757959777746319698979526182946558649778907703714686717365894911e-7
comment: $\Delta(\mathbb{Z}^{18})=\frac{\pi^{9}}{95126814720}$
Z
18
hermite:
1
comment: $\gamma(\mathbb{Z}^{18})=1$
equals: One
Z
19
centre:
1/524288
comment: $\delta(\mathbb{Z}^{19})=\frac{1}{524288}$; $\det=1$, $\mu=1$, kissing number $38$
Z
19
density:
8.892364698426922946892373240782258200852742081204737396017330010417444312364870218911243313745319834e-8
comment: $\Delta(\mathbb{Z}^{19})=\frac{\pi^{9}}{335221286400}$
Z
19
hermite:
1
comment: $\gamma(\mathbb{Z}^{19})=1$
equals: One
Z
2
centre:
1/4
comment: $\delta(\mathbb{Z}^{2})=\frac{1}{4}$; $\det=1$, $\mu=1$, kissing number $4$
Z
2
density:
0.7853981633974483096156608458198757210492923498437764552437361480769541015715522496570087063355292670
comment: $\Delta(\mathbb{Z}^{2})=\frac{\pi}{4}$
equals: Rational_multiples_of_pi#1/4
Z
2
hermite:
1
comment: $\gamma(\mathbb{Z}^{2})=1$
equals: One
Z
20
centre:
1/1048576
comment: $\delta(\mathbb{Z}^{20})=\frac{1}{1048576}$; $\det=1$, $\mu=1$, kissing number $40$
Z
20
density:
2.461136950494199754009084153451809850424427132429760550670850040241898473353171907000421357463503371e-8
comment: $\Delta(\mathbb{Z}^{20})=\frac{\pi^{10}}{3805072588800}$
Z
20
hermite:
1
comment: $\gamma(\mathbb{Z}^{20})=1$
equals: One
Z
21
centre:
1/2097152
comment: $\delta(\mathbb{Z}^{21})=\frac{1}{2097152}$; $\det=1$, $\mu=1$, kissing number $42$
Z
21
density:
6.651473240385532942959593103722605994930364024042697754207189229234874860935698938715757476073076848e-9
comment: $\Delta(\mathbb{Z}^{21})=\frac{\pi^{10}}{14079294028800}$
Z
21
hermite:
1
comment: $\gamma(\mathbb{Z}^{21})=1$
equals: One
Z
22
centre:
1/4194304
comment: $\delta(\mathbb{Z}^{22})=\frac{1}{4194304}$; $\det=1$, $\mu=1$, kissing number $44$
Z
22
density:
1.757247673443401045145771493265815139105710562710999834513626745657322448688053597757085883677264729e-9
comment: $\Delta(\mathbb{Z}^{22})=\frac{\pi^{11}}{167423193907200}$
Z
22
hermite:
1
comment: $\gamma(\mathbb{Z}^{22})=1$
equals: One
Z
23
centre:
1/8388608
comment: $\delta(\mathbb{Z}^{23})=\frac{1}{8388608}$; $\det=1$, $\mu=1$, kissing number $46$
Z
23
density:
4.542656405987888505236069934137664087680615958066643498562855788039774194377159920561126620315227229e-10
comment: $\Delta(\mathbb{Z}^{23})=\frac{\pi^{11}}{647647525324800}$
Z
23
hermite:
1
comment: $\gamma(\mathbb{Z}^{23})=1$
equals: One
Z
24
centre:
1/16777216
comment: $\delta(\mathbb{Z}^{24})=\frac{1}{16777216}$; $\det=1$, $\mu=1$, kissing number $48$
Z
24
density:
1.150115912797405152263251282981608891587538608908834071299223671753105899245134312800972281805362650e-10
comment: $\Delta(\mathbb{Z}^{24})=\frac{\pi^{12}}{8036313307545600}$
Z
24
hermite:
1
comment: $\gamma(\mathbb{Z}^{24})=1$
equals: One
Z
3
centre:
1/8
comment: $\delta(\mathbb{Z}^{3})=\frac{1}{8}$; $\det=1$, $\mu=1$, kissing number $6$
Z
3
density:
0.5235987755982988730771072305465838140328615665625176368291574320513027343810348331046724708903528447
comment: $\Delta(\mathbb{Z}^{3})=\frac{\pi}{6}$
equals: Rational_multiples_of_pi#1/6
Z
3
hermite:
1
comment: $\gamma(\mathbb{Z}^{3})=1$
equals: One
Z
4
centre:
1/16
comment: $\delta(\mathbb{Z}^{4})=\frac{1}{16}$; $\det=1$, $\mu=1$, kissing number $8$
Z
4
density:
0.3084251375340424568385778437461297229785531064762747070754171680068764007006001638438054188662047572
comment: $\Delta(\mathbb{Z}^{4})=\frac{\pi^{2}}{32}$
Z
4
hermite:
1
comment: $\gamma(\mathbb{Z}^{4})=1$
equals: One
Z
5
centre:
1/32
comment: $\delta(\mathbb{Z}^{5})=\frac{1}{32}$; $\det=1$, $\mu=1$, kissing number $10$
Z
5
density:
0.1644934066848226436472415166646025189218949901206798437735558229370007470403200873833628900619758705
comment: $\Delta(\mathbb{Z}^{5})=\frac{\pi^{2}}{60}$
Z
5
hermite:
1
comment: $\gamma(\mathbb{Z}^{5})=1$
equals: One
Z
6
centre:
1/64
comment: $\delta(\mathbb{Z}^{6})=\frac{1}{64}$; $\det=1$, $\mu=1$, kissing number $12$
Z
6
density:
0.08074551218828078170696957048724321667246168897365913462016806797866248676423360947282995060573668279
comment: $\Delta(\mathbb{Z}^{6})=\frac{\pi^{3}}{384}$
Z
6
hermite:
1
comment: $\gamma(\mathbb{Z}^{6})=1$
equals: One
Z
7
centre:
1/128
comment: $\delta(\mathbb{Z}^{7})=\frac{1}{128}$; $\det=1$, $\mu=1$, kissing number $14$
Z
7
density:
0.03691223414321407163747180365131118476455391495938703296921968821881713680650679290186512027690819785
comment: $\Delta(\mathbb{Z}^{7})=\frac{\pi^{3}}{840}$
Z
7
hermite:
1
comment: $\gamma(\mathbb{Z}^{7})=1$
equals: One
Z
8
centre:
1/256
comment: $\delta(\mathbb{Z}^{8})=\frac{1}{256}$; $\det=1$, $\mu=1$, kissing number $16$
Z
8
density:
0.01585434424381550085228521039855226420080201589724697618676234690591973558867630146323598099523974488
comment: $\Delta(\mathbb{Z}^{8})=\frac{\pi^{4}}{6144}$
Z
8
hermite:
1
comment: $\gamma(\mathbb{Z}^{8})=1$
equals: One
Z
9
centre:
1/512
comment: $\delta(\mathbb{Z}^{9})=\frac{1}{512}$; $\det=1$, $\mu=1$, kissing number $18$
Z
9
density:
0.006442400200661536854261926765125999421278279475706707783827239377643575096351005039029224023462499507
comment: $\Delta(\mathbb{Z}^{9})=\frac{\pi^{4}}{15120}$
Z
9
hermite:
1
comment: $\gamma(\mathbb{Z}^{9})=1$
equals: One
Definition
For a lattice $L\subset\mathbb{R}^n$ with minimal norm $\mu$, packing radius $\rho=\sqrt{\mu}/2$ and determinant $\det L$, the packing density $\Delta(L)=V_n\rho^n/\sqrt{\det L}$ of the sphere packing [20] with centres at the points of $L$, its centre density $\delta(L)=\rho^n/\sqrt{\det L}$ and its Hermite number $\gamma(L)=\mu/(\det L)^{1/n}$ [12], with $V_n$ the volume of the unit ball, for $\mathbb{Z}^n$, the root lattices, their duals, the laminated lattices $\Lambda_n$ and the Coxeter–Todd lattice $K_{12}$.
Parameters
family
—   family of the lattice
$n$
—   dimension ($n\geq 1$ for $\mathbb{Z}^n$ and $A_n$; $n\geq 4$ for $D_n$; $n\in\{6,7,8\}$ for $E_n$; $n\geq 3$ for $A_n^{*}$; $n\geq 5$ for $D_n^{*}$; $n\in\{6,7\}$ for $E_n^{*}$; $n\geq 9$ for $\Lambda_n$; $n=12$ for $K_n$)
Formulas
(1)
$\gamma(A_n^{*})=n\,(n+1)^{1/n-1}$ for $n\geq 2$, $\gamma(D_n^{*})=4^{1/n}$ for $n\geq 4$, $\gamma(E_6^{*})=\frac{4}{3}\cdot 3^{1/6}$ and $\gamma(E_7^{*})=\frac{3}{2}\cdot 2^{1/7}$, from $\det L^{*}=1/\det L$ and the minimal norms $n/(n+1)$, $1$, $4/3$ and $3/2$ when the root lattice has $\mu=2$ [12], chapter 4.
(2)
$\gamma_n=\max_L\gamma(L)$ satisfies $\gamma_n^{\,n}=1,\ \frac{4}{3},\ 2,\ 4,\ 8,\ \frac{64}{3},\ 64,\ 256$ for $n=1,\ldots,8$, attained by $\mathbb{Z}$, $A_2$, $A_3$, $D_4$, $D_5$, $E_6$, $E_7$, $E_8$, and $\gamma_{24}=4$, attained by $\Lambda_{24}$ [4]; these are the only dimensions in which $\gamma_n$ is known [21].
(3)
With $\Lambda_n$ scaled to $\mu=4$, $\delta(\Lambda_n)=1/\sqrt{\det\Lambda_n}$ and $\gamma(\Lambda_n)=4\,(\det\Lambda_n)^{-1/n}$, where $\det\Lambda_n$ for $n=0,1,\ldots,24$ is $1$, $4$, $12$, $32$, $64$, $128$, $192$, $256$, $256$, $512$, $768$, $1024$, $1024$, $1024$, $768$, $512$, $256$, $256$, $192$, $128$, $64$, $32$, $12$, $4$, $1$ [18]; so $\det\Lambda_{24-n}=\det\Lambda_n$ and $\delta(\Lambda_{24-n})=\delta(\Lambda_n)$ for $0\leq n\leq 24$. $\delta(K_{12})=1/27$ and $\gamma(K_{12})=4/\sqrt{3}$.
(4)
$\delta(L)=\bigl(\gamma(L)/4\bigr)^{n/2}$ and $\Delta(L)=V_n\,\delta(L)$, where $V_n=\pi^{n/2}/\Gamma(n/2+1)$ is the volume of the unit ball; in particular $\Delta(\mathbb{Z}^n)=V_n/2^n$ and $\gamma(\mathbb{Z}^n)=1$.
(5)
$\gamma(A_n)=2\,(n+1)^{-1/n}$, $\gamma(D_n)=2\cdot 4^{-1/n}$, $\gamma(E_6)=2\cdot 3^{-1/6}$, $\gamma(E_7)=2^{6/7}$, $\gamma(E_8)=2$, from $\mu=2$ and the determinants $n+1$, $4$, $3$, $2$, $1$.
Comments
(6)
The minimal norm is $\mu=\min\{v\cdot v : v\in L,\ v\neq 0\}$, the determinant $\det L$ is the determinant of a Gram matrix of $L$, the square of the volume of a fundamental domain, and $V_n=\pi^{n/2}/\Gamma(n/2+1)$. The three quantities are those of Conway and Sloane [12], chapter 1, and all three are unchanged when $L$ is scaled, so no scaling of a named lattice has to be chosen. $\gamma(L)$ is also called the Hermite invariant of $L$, and some authors write $\gamma$ for its square root, the ratio of the length of a shortest vector to the $n$-th root of the covolume. Its supremum over all lattices in $\mathbb{R}^n$ is Hermite's constant $\gamma_n$ [21], known for $n\leq 8$ and $n=24$ and attained by lattices listed here, (2). The densest lattice packing in $\mathbb{R}^n$ is the lattice with the largest $\gamma(L)$, equivalently the largest $\Delta(L)$ or $\delta(L)$.
(7)
The comment on each entry gives its value in closed form, always $\pi^{\lfloor n/2\rfloor}$ times an algebraic number of degree at most $n$; on the centre density, the determinant, the minimal norm and the kissing number of the lattice, its number of shortest vectors, which for $\Lambda_n$ and $K_{12}$ are OEIS A002336 [19]; and, where it is known, whether the lattice is the densest lattice packing of its dimension, the densest packing of any kind, or the densest lattice packing known, the last according to the catalogue's table of densest packings [16], last revised in February 2012. The densest lattice packing is known for $n\leq 8$, $\Lambda_n$ in each case (Lagrange for $n=2$ [9], Gauss for $n=3$ [6], Korkine and Zolotareff for $n=4,5$ [8], Blichfeldt for $n=6,7,8$ [2]), and for $n=24$, the Leech lattice [4]; the densest packing of any kind is known for $n=1$, $2$ [13], $3$ [7], $8$ [14] and $24$ [3], and is the lattice packing in each case. In dimensions $10$, $11$, $13$, $18$, $20$ and $22$ a nonlattice packing denser than every lattice packing known exists [16].
(8)
Each lattice is given by an integral Gram matrix, and every value is a function of its dimension, determinant and minimal norm alone: the identity matrix for $\mathbb{Z}^n$, the Cartan matrix for a root lattice, so that $\mu=2$ and $\det A_n=n+1$, $\det D_n=4$, $\det E_6=3$, $\det E_7=2$, $\det E_8=1$; the inverse Gram matrix for a dual, so that $\det L^{*}=1/\det L$; and the Gram matrices of the Catalogue of Lattices [15] for $\Lambda_9$ to $\Lambda_{24}$ and $K_{12}$, scaled to $\mu=4$, whose determinants $\det\Lambda_n$ are OEIS A028921 [18] and $\det K_{12}=729$.
(9)
$\mathbb{Z}^n$ is the cubic lattice; $A_n$, $D_n$ and $E_n$ are the root lattices [12], chapter 4, with $A_2$ the hexagonal lattice, $A_3=D_3$ the face-centred cubic lattice and $A_3^{*}$ the body-centred cubic lattice; $\Lambda_n$ are the laminated lattices of Conway and Sloane [10], with $\Lambda_{16}$ the Barnes–Wall lattice $BW_{16}$ [1] and $\Lambda_{24}$ the Leech lattice [11]; $K_{12}$ is the Coxeter–Todd lattice [5]. A lattice with two names is listed once: $D_3$ under $A_3$; $\Lambda_n$ for $n\leq 8$, which is $\mathbb{Z}$, $A_2$, $A_3$, $D_4$, $D_5$, $E_6$, $E_7$, $E_8$, under those names; and $A_1^{*}$, $A_2^{*}$ and $D_4^{*}$ not at all, being similar to $A_1$, $A_2$ and $D_4$. $E_8^{*}=E_8$. Every family stops at $n=24$, where the laminated sequence and the proven optimality results end. The lattices $\Lambda_{11}$, $\Lambda_{12}$ and $\Lambda_{13}$ are not the densest lattices known in their dimensions: $K_{12}$ and its laminations $K_{11}$ and $K_{13}$, which are not listed, are denser [16].
Programs
(P1)
Sage
G = CartanMatrix(['E', 8]); n = G.nrows()
det = G.det(); mu = IntegralLattice(G).minimum()      # 1, 2
hermite = mu / det^(1/n)                               # 2
delta = (sqrt(mu)/2)^n / sqrt(det)                     # 1/16
Delta = pi^(n/2) / gamma(n/2 + 1) * delta              # pi^4/384
pari(G).qfminim()[0]                                   # 240, the kissing number
References
[1]
E. S. Barnes and G. E. Wall, Some extreme forms defined in terms of Abelian groups, Journal of the Australian Mathematical Society 1 (1959), 47–63.
[2]
H. F. Blichfeldt, The minimum values of positive quadratic forms in six, seven and eight variables, Mathematische Zeitschrift 39 (1935), 1–15.
[3]
H. Cohn, A. Kumar, S. D. Miller, D. Radchenko and M. Viazovska, The sphere packing problem in dimension 24, Annals of Mathematics 185 (2017), 1017–1033.
[4]
H. Cohn and A. Kumar, Optimality and uniqueness of the Leech lattice among lattices, Annals of Mathematics 170 (2009), 1003–1050.
[5]
H. S. M. Coxeter and J. A. Todd, An extreme duodenary form, Canadian Journal of Mathematics 5 (1953), 384–392.
[6]
C. F. Gauss, Untersuchungen über die Eigenschaften der positiven ternären quadratischen Formen von Ludwig August Seeber, Göttingische gelehrte Anzeigen (1831); Werke II, 188–196.
[7]
T. C. Hales, A proof of the Kepler conjecture, Annals of Mathematics 162 (2005), 1065–1185.
[8]
A. Korkine and G. Zolotareff, Sur les formes quadratiques positives quaternaires, Mathematische Annalen 5 (1872), 581–583; Sur les formes quadratiques positives, Mathematische Annalen 11 (1877), 242–292.
[9]
J. L. Lagrange, Recherches d'arithmétique, Nouveaux Mémoires de l'Académie royale des Sciences et Belles-Lettres de Berlin (1773), 265–312.
[10]
J. H. Conway and N. J. A. Sloane, Laminated lattices, Annals of Mathematics 116 (1982), 593–620.
[11]
J. Leech, Notes on sphere packings, Canadian Journal of Mathematics 19 (1967), 251–267.
[12]
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, third edition, Grundlehren der mathematischen Wissenschaften 290, Springer, 1999.
[13]
A. Thue, Über die dichteste Zusammenstellung von kongruenten Kreisen in einer Ebene, Norske Videnskabs-Selskabets Skrifter 1 (1910), 1–9; L. Fejes Tóth, Über einen geometrischen Satz, Mathematische Zeitschrift 46 (1940), 83–85.
[14]
M. S. Viazovska, The sphere packing problem in dimension 8, Annals of Mathematics 185 (2017), 991–1015.
Links
Similar tables
Volume of the $d$-dimensional unit ball —   $V_n$, the factor by which $\Delta(L)$ and $\delta(L)$ differ
Rational multiples of pi —   holds $\Delta(\mathbb{Z}^2)=\pi/4$ and $\Delta(\mathbb{Z}^3)=\pi/6$
Algebraic numbers of degree 2 —   holds $\gamma(A_2)=2/\sqrt{3}$ and $\gamma(D_4)=\sqrt{2}$
Data properties
Table is complete: no (every family to $n=24$ is here: $\mathbb{Z}^n$ and $A_n$ for $1\leq n\leq 24$, $D_n$ for $4\leq n\leq 24$, $E_6$, $E_7$, $E_8$, $A_n^{*}$ for $3\leq n\leq 24$, $D_n^{*}$ for $5\leq n\leq 24$, $E_6^{*}$, $E_7^{*}$, $\Lambda_n$ for $9\leq n\leq 24$ and $K_{12}$, 133 lattices and 399 entries)
How well the digits are known: proven
How they were obtained:

For each lattice the determinant of its integral Gram matrix is an exact integer and the minimal norm and the number of shortest vectors come from PARI's qfminim, so $\delta^2=(\mu/4)^n/\det L$ and $\gamma^n=\mu^n/\det L$ are exact rationals; a value that is rational is written exactly, and the rest are balls in arb, a square root, an $n$-th root, and for $\Delta$ the product with $\pi^{\lfloor n/2\rfloor}$ and the exact rational $V_n/\pi^{\lfloor n/2\rfloor}$, computed with 64 guard bits beyond the 100 digits written; the widest ball relative to its value, $\Delta(\Lambda_{22})$, has radius $4.5\cdot 10^{-119}$.

more

Before a lattice is accepted its kissing number and determinant must equal the known values ($2n$, $n(n+1)$, $2n(n-1)$, $72$, $126$, $240$ for $\mathbb{Z}^n$, $A_n$, $D_n$, $E_6$, $E_7$, $E_8$; $2(n+1)$, $2n$, $54$, $56$ for the duals; the catalogue's lines for $\Lambda_n$ and $K_{12}$), and $\delta^2=(\gamma/4)^n$ must hold exactly. Outside the generator, the Gram matrices transcribed from the catalogue were compared entry by entry with the catalogue pages fetched again, and their determinants, minimal norms and kissing numbers with the pages' DET, MINIMAL_NORM and KISSING_NUMBER lines; $\Lambda_{16}$ built from the Reed–Muller code $RM(1,4)$ was found isometric to the catalogue's Gram matrix by PARI's qfisom, and the catalogue's Gram matrix of $\Lambda_{24}$, like the lattice built from the extended Golay code, is even, unimodular and without vectors of norm $2$, hence the Leech lattice by Conway's uniqueness theorem; the Cartan matrices were compared with Sage's; $\Delta(A_2)$, $\Delta(A_3)$, $\Delta(D_4)$, $\Delta(D_5)$, $\Delta(E_6)$, $\Delta(E_7)$, $\Delta(E_8)$, $\Delta(\Lambda_{24})$, $\delta(D_5)$, $\delta(E_6)$, $\gamma(E_6)$ and $\gamma(E_7)$ agree with OEIS A093766, A093825, A222068, A222069, A222070, A222071, A222072, A260646, A222066, A222067, A246184 and A246722 to every digit those entries give; eleven of the fourteen HERMITE_NUMBER lines and both DENSITY lines of the catalogue pages agree to their twelve digits, and the lines of $\Lambda_{13}$, $\Lambda_{20}$ and $\Lambda_{22}$ to eight, those three being off in their ninth or tenth digit; the determinants of $\Lambda_n$ at minimal norm $4$ are OEIS A028921 and the kissing numbers OEIS A002336 for $1\leq n\leq 24$; the largest $\gamma^n$ in each dimension $n\leq 8$ is OEIS A007361/A007362 and the largest $\delta$ in each dimension $n\leq 24$ is the catalogue's table of densest packings; $\Delta=V_n\delta$ holds in balls against the stored digits of the table of unit-ball volumes and against $V_n$ from arb's Gamma function; the closed forms of (5), (1) and (3) hold in balls; the entries linked to other tables agree with the stored digits there; and the closed form in every entry's comment, evaluated in balls, encloses the entry. Controls that must fail did: the Gram matrix of $\Lambda_{12}$ offered as $K_{12}$ is refused, and $\Delta(D_4)$ moved by $10^{-50}$ no longer matches A222068.

Entries are of type: real number


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