History of Packing densities and Hermite numbers of the classical lattices

 Display properties:   number-header: $\Delta(L)$, $\delta(L)$ or $\gamma(L)$-Numbers: []+Numbers:+- params:+    family: Z+    n: '1'+    expression: density+  number: '1'+  comment: $\Delta(\mathbb{Z})=1$; the densest packing of any kind in dimension 1;+    $\mathbb{Z}=\Lambda_{1}$+  equals: HREF{One}+- params:+    family: Z+    n: '1'+    expression: centre+  number: 1/2+  comment: $\delta(\mathbb{Z})=\frac{1}{2}$; $\det=1$, $\mu=1$, kissing number $2$;+    $\mathbb{Z}=\Lambda_{1}$+- params:+    family: Z+    n: '1'+    expression: hermite+  number: '1'+  comment: $\gamma(\mathbb{Z})=1$; Hermite's constant $\gamma_{1}$; $\mathbb{Z}=\Lambda_{1}$+  equals: HREF{One}+- params:+    family: Z+    n: '2'+    expression: density+  number: '0.7853981633974483096156608458198757210492923498437764552437361480769541015715522496570087063355292670'+  comment: $\Delta(\mathbb{Z}^{2})=\frac{\pi}{4}$+  equals: HREF{Rational_multiples_of_pi#1/4}+- params:+    family: Z+    n: '2'+    expression: centre+  number: 1/4+  comment: $\delta(\mathbb{Z}^{2})=\frac{1}{4}$; $\det=1$, $\mu=1$, kissing number+    $4$+- params:+    family: Z+    n: '2'+    expression: hermite+  number: '1'+  comment: $\gamma(\mathbb{Z}^{2})=1$+  equals: HREF{One}+- params:+    family: Z+    n: '3'+    expression: density+  number: '0.5235987755982988730771072305465838140328615665625176368291574320513027343810348331046724708903528447'+  comment: $\Delta(\mathbb{Z}^{3})=\frac{\pi}{6}$+  equals: HREF{Rational_multiples_of_pi#1/6}+- params:+    family: Z+    n: '3'+    expression: centre+  number: 1/8+  comment: $\delta(\mathbb{Z}^{3})=\frac{1}{8}$; $\det=1$, $\mu=1$, kissing number+    $6$+- params:+    family: Z+    n: '3'+    expression: hermite+  number: '1'+  comment: $\gamma(\mathbb{Z}^{3})=1$+  equals: HREF{One}+- params:+    family: Z+    n: '4'+    expression: density+  number: '0.3084251375340424568385778437461297229785531064762747070754171680068764007006001638438054188662047572'+  comment: $\Delta(\mathbb{Z}^{4})=\frac{\pi^{2}}{32}$+- params:+    family: Z+    n: '4'+    expression: centre+  number: 1/16+  comment: $\delta(\mathbb{Z}^{4})=\frac{1}{16}$; $\det=1$, $\mu=1$, kissing number+    $8$+- params:+    family: Z+    n: '4'+    expression: hermite+  number: '1'+  comment: $\gamma(\mathbb{Z}^{4})=1$+  equals: HREF{One}+- params:+    family: Z+    n: '5'+    expression: density+  number: '0.1644934066848226436472415166646025189218949901206798437735558229370007470403200873833628900619758705'+  comment: $\Delta(\mathbb{Z}^{5})=\frac{\pi^{2}}{60}$+- params:+    family: Z+    n: '5'+    expression: centre+  number: 1/32+  comment: $\delta(\mathbb{Z}^{5})=\frac{1}{32}$; $\det=1$, $\mu=1$, kissing number+    $10$+- params:+    family: Z+    n: '5'+    expression: hermite+  number: '1'+  comment: $\gamma(\mathbb{Z}^{5})=1$+  equals: HREF{One}+- params:+    family: Z+    n: '6'+    expression: density+  number: '0.08074551218828078170696957048724321667246168897365913462016806797866248676423360947282995060573668279'+  comment: $\Delta(\mathbb{Z}^{6})=\frac{\pi^{3}}{384}$+- params:+    family: Z+    n: '6'+    expression: centre+  number: 1/64+  comment: $\delta(\mathbb{Z}^{6})=\frac{1}{64}$; $\det=1$, $\mu=1$, kissing number+    $12$+- params:+    family: Z+    n: '6'+    expression: hermite+  number: '1'+  comment: $\gamma(\mathbb{Z}^{6})=1$+  equals: HREF{One}+- params:+    family: Z+    n: '7'+    expression: density+  number: '0.03691223414321407163747180365131118476455391495938703296921968821881713680650679290186512027690819785'+  comment: $\Delta(\mathbb{Z}^{7})=\frac{\pi^{3}}{840}$+- params:+    family: Z+    n: '7'+    expression: centre+  number: 1/128+  comment: $\delta(\mathbb{Z}^{7})=\frac{1}{128}$; $\det=1$, $\mu=1$, kissing number+    $14$+- params:+    family: Z+    n: '7'+    expression: hermite+  number: '1'+  comment: $\gamma(\mathbb{Z}^{7})=1$+  equals: HREF{One}+- params:+    family: Z+    n: '8'+    expression: density+  number: '0.01585434424381550085228521039855226420080201589724697618676234690591973558867630146323598099523974488'+  comment: $\Delta(\mathbb{Z}^{8})=\frac{\pi^{4}}{6144}$+- params:+    family: Z+    n: '8'+    expression: centre+  number: 1/256+  comment: $\delta(\mathbb{Z}^{8})=\frac{1}{256}$; $\det=1$, $\mu=1$, kissing number+    $16$+- params:+    family: Z+    n: '8'+    expression: hermite+  number: '1'+  comment: $\gamma(\mathbb{Z}^{8})=1$+  equals: HREF{One}+- params:+    family: Z+    n: '9'+    expression: density+  number: '0.006442400200661536854261926765125999421278279475706707783827239377643575096351005039029224023462499507'+  comment: $\Delta(\mathbb{Z}^{9})=\frac{\pi^{4}}{15120}$+- params:+    family: Z+    n: '9'+    expression: centre+  number: 1/512+  comment: $\delta(\mathbb{Z}^{9})=\frac{1}{512}$; $\det=1$, $\mu=1$, kissing number+    $18$+- params:+    family: Z+    n: '9'+    expression: hermite+  number: '1'+  comment: $\gamma(\mathbb{Z}^{9})=1$+  equals: HREF{One}+- params:+    family: Z+    n: '10'+    expression: density+  number: '0.002490394570192720160015798421577438203778488823470662358019958037811157162545687048224149580676311189'+  comment: $\Delta(\mathbb{Z}^{10})=\frac{\pi^{5}}{122880}$+- params:+    family: Z+    n: '10'+    expression: centre+  number: 1/1024+  comment: $\delta(\mathbb{Z}^{10})=\frac{1}{1024}$; $\det=1$, $\mu=1$, kissing number+    $20$+- params:+    family: Z+    n: '10'+    expression: hermite+  number: '1'+  comment: $\gamma(\mathbb{Z}^{10})=1$+  equals: HREF{One}+- params:+    family: Z+    n: '11'+    expression: density+  number: '0.0009199725973583497272208432841613624533438573431579935983450350038667478118494890105993972476957224594'+  comment: $\Delta(\mathbb{Z}^{11})=\frac{\pi^{5}}{332640}$+- params:+    family: Z+    n: '11'+    expression: centre+  number: 1/2048+  comment: $\delta(\mathbb{Z}^{11})=\frac{1}{2048}$; $\det=1$, $\mu=1$, kissing number+    $22$+- params:+    family: Z+    n: '11'+    expression: hermite+  number: '1'+  comment: $\gamma(\mathbb{Z}^{11})=1$+  equals: HREF{One}+- params:+    family: Z+    n: '12'+    expression: density+  number: '0.0003259918869273900136414318317506306623220545546064746825972073660902553686667040383230324331162908719'+  comment: $\Delta(\mathbb{Z}^{12})=\frac{\pi^{6}}{2949120}$+- params:+    family: Z+    n: '12'+    expression: centre+  number: 1/4096+  comment: $\delta(\mathbb{Z}^{12})=\frac{1}{4096}$; $\det=1$, $\mu=1$, kissing number+    $24$+- params:+    family: Z+    n: '12'+    expression: hermite+  number: '1'+  comment: $\gamma(\mathbb{Z}^{12})=1$+  equals: HREF{One}+- params:+    family: Z+    n: '13'+    expression: density+  number: '0.0001111607366678812434128625360348470856536076802920513070194939536718020304744272178630653384985287555'+  comment: $\Delta(\mathbb{Z}^{13})=\frac{\pi^{6}}{8648640}$+- params:+    family: Z+    n: '13'+    expression: centre+  number: 1/8192+  comment: $\delta(\mathbb{Z}^{13})=\frac{1}{8192}$; $\det=1$, $\mu=1$, kissing number+    $26$+- params:+    family: Z+    n: '13'+    expression: hermite+  number: '1'+  comment: $\gamma(\mathbb{Z}^{13})=1$+  equals: HREF{One}+- params:+    family: Z+    n: '14'+    expression: density+  number: '0.00003657620418217725078660518698401123327730490639453951230432703523645715461965423924205056596895259961'+  comment: $\Delta(\mathbb{Z}^{14})=\frac{\pi^{7}}{82575360}$+- params:+    family: Z+    n: '14'+    expression: centre+  number: 1/16384+  comment: $\delta(\mathbb{Z}^{14})=\frac{1}{16384}$; $\det=1$, $\mu=1$, kissing number+    $28$+- params:+    family: Z+    n: '14'+    expression: hermite+  number: '1'+  comment: $\gamma(\mathbb{Z}^{14})=1$+  equals: HREF{One}+- params:+    family: Z+    n: '15'+    expression: density+  number: '0.00001164072512278150887505321257859440648825492592012694968131496008768675255028001273779635728118336037'+  comment: $\Delta(\mathbb{Z}^{15})=\frac{\pi^{7}}{259459200}$+- params:+    family: Z+    n: '15'+    expression: centre+  number: 1/32768+  comment: $\delta(\mathbb{Z}^{15})=\frac{1}{32768}$; $\det=1$, $\mu=1$, kissing number+    $30$+- params:+    family: Z+    n: '15'+    expression: hermite+  number: '1'+  comment: $\gamma(\mathbb{Z}^{15})=1$+  equals: HREF{One}+- params:+    family: Z+    n: '16'+    expression: density+  number: '0.000003590860448591510079069203991324850792618849720894754005655444393107591697840984395963980783363718883'+  comment: $\Delta(\mathbb{Z}^{16})=\frac{\pi^{8}}{2642411520}$+- params:+    family: Z+    n: '16'+    expression: centre+  number: 1/65536+  comment: $\delta(\mathbb{Z}^{16})=\frac{1}{65536}$; $\det=1$, $\mu=1$, kissing number+    $32$+- params:+    family: Z+    n: '16'+    expression: hermite+  number: '1'+  comment: $\gamma(\mathbb{Z}^{16})=1$+  equals: HREF{One}+- params:+    family: Z+    n: '17'+    expression: density+  number: '0.000001075600486123192122774712522398032001211522168785404262144683046513547829012782820850566500381757305'+  comment: $\Delta(\mathbb{Z}^{17})=\frac{\pi^{8}}{8821612800}$+- params:+    family: Z+    n: '17'+    expression: centre+  number: 1/131072+  comment: $\delta(\mathbb{Z}^{17})=\frac{1}{131072}$; $\det=1$, $\mu=1$, kissing+    number $34$+- params:+    family: Z+    n: '17'+    expression: hermite+  number: '1'+  comment: $\gamma(\mathbb{Z}^{17})=1$+  equals: HREF{One}+- params:+    family: Z+    n: '18'+    expression: density+  number: '3.133616890378121520950407620380338470757959777746319698979526182946558649778907703714686717365894911e-7'+  comment: $\Delta(\mathbb{Z}^{18})=\frac{\pi^{9}}{95126814720}$+- params:+    family: Z+    n: '18'+    expression: centre+  number: 1/262144+  comment: $\delta(\mathbb{Z}^{18})=\frac{1}{262144}$; $\det=1$, $\mu=1$, kissing+    number $36$+- params:+    family: Z+    n: '18'+    expression: hermite+  number: '1'+  comment: $\gamma(\mathbb{Z}^{18})=1$+  equals: HREF{One}+- params:+    family: Z+    n: '19'+    expression: density+  number: '8.892364698426922946892373240782258200852742081204737396017330010417444312364870218911243313745319834e-8'+  comment: $\Delta(\mathbb{Z}^{19})=\frac{\pi^{9}}{335221286400}$+- params:+    family: Z+    n: '19'+    expression: centre+  number: 1/524288+  comment: $\delta(\mathbb{Z}^{19})=\frac{1}{524288}$; $\det=1$, $\mu=1$, kissing+    number $38$+- params:+    family: Z+    n: '19'+    expression: hermite+  number: '1'+  comment: $\gamma(\mathbb{Z}^{19})=1$+  equals: HREF{One}+- params:+    family: Z+    n: '20'+    expression: density+  number: '2.461136950494199754009084153451809850424427132429760550670850040241898473353171907000421357463503371e-8'+  comment: $\Delta(\mathbb{Z}^{20})=\frac{\pi^{10}}{3805072588800}$+- params:+    family: Z+    n: '20'+    expression: centre+  number: 1/1048576+  comment: $\delta(\mathbb{Z}^{20})=\frac{1}{1048576}$; $\det=1$, $\mu=1$, kissing+    number $40$+- params:+    family: Z+    n: '20'+    expression: hermite+  number: '1'+  comment: $\gamma(\mathbb{Z}^{20})=1$+  equals: HREF{One}+- params:+    family: Z+    n: '21'+    expression: density+  number: '6.651473240385532942959593103722605994930364024042697754207189229234874860935698938715757476073076848e-9'+  comment: $\Delta(\mathbb{Z}^{21})=\frac{\pi^{10}}{14079294028800}$+- params:+    family: Z+    n: '21'+    expression: centre+  number: 1/2097152+  comment: $\delta(\mathbb{Z}^{21})=\frac{1}{2097152}$; $\det=1$, $\mu=1$, kissing+    number $42$+- params:+    family: Z+    n: '21'+    expression: hermite+  number: '1'+  comment: $\gamma(\mathbb{Z}^{21})=1$+  equals: HREF{One}+- params:+    family: Z+    n: '22'+    expression: density+  number: '1.757247673443401045145771493265815139105710562710999834513626745657322448688053597757085883677264729e-9'+  comment: $\Delta(\mathbb{Z}^{22})=\frac{\pi^{11}}{167423193907200}$+- params:+    family: Z+    n: '22'+    expression: centre+  number: 1/4194304+  comment: $\delta(\mathbb{Z}^{22})=\frac{1}{4194304}$; $\det=1$, $\mu=1$, kissing+    number $44$+- params:+    family: Z+    n: '22'+    expression: hermite+  number: '1'+  comment: $\gamma(\mathbb{Z}^{22})=1$+  equals: HREF{One}+- params:+    family: Z+    n: '23'+    expression: density+  number: '4.542656405987888505236069934137664087680615958066643498562855788039774194377159920561126620315227229e-10'+  comment: $\Delta(\mathbb{Z}^{23})=\frac{\pi^{11}}{647647525324800}$+- params:+    family: Z+    n: '23'+    expression: centre+  number: 1/8388608+  comment: $\delta(\mathbb{Z}^{23})=\frac{1}{8388608}$; $\det=1$, $\mu=1$, kissing+    number $46$+- params:+    family: Z+    n: '23'+    expression: hermite+  number: '1'+  comment: $\gamma(\mathbb{Z}^{23})=1$+  equals: HREF{One}+- params:+    family: Z+    n: '24'+    expression: density+  number: '1.150115912797405152263251282981608891587538608908834071299223671753105899245134312800972281805362650e-10'+  comment: $\Delta(\mathbb{Z}^{24})=\frac{\pi^{12}}{8036313307545600}$+- params:+    family: Z+    n: '24'+    expression: centre+  number: 1/16777216+  comment: $\delta(\mathbb{Z}^{24})=\frac{1}{16777216}$; $\det=1$, $\mu=1$, kissing+    number $48$+- params:+    family: Z+    n: '24'+    expression: hermite+  number: '1'+  comment: $\gamma(\mathbb{Z}^{24})=1$+  equals: HREF{One}+- params:+    family: A+    n: '1'+    expression: density+  number: '1'+  comment: $\Delta(A_{1})=1$; the densest packing of any kind in dimension 1; $A_1=\sqrt{2}\,\mathbb{Z}$+  equals: HREF{One}+- params:+    family: A+    n: '1'+    expression: centre+  number: 1/2+  comment: $\delta(A_{1})=\frac{1}{2}$; $\det=2$, $\mu=2$, kissing number $2$; $A_1=\sqrt{2}\,\mathbb{Z}$+- params:+    family: A+    n: '1'+    expression: hermite+  number: '1'+  comment: $\gamma(A_{1})=1$; Hermite's constant $\gamma_{1}$; $A_1=\sqrt{2}\,\mathbb{Z}$+  equals: HREF{One}+- params:+    family: A+    n: '2'+    expression: density+  number: '0.9068996821171089252970391288210778661420331240463702877849424676940615905631769418420624941060300844'+  comment: $\Delta(A_{2})=\frac{\sqrt{3}\,\pi}{6}$; the densest packing of any kind+    in dimension 2 CITE{ThueFejesToth}; $A_{2}=\Lambda_{2}$+- params:+    family: A+    n: '2'+    expression: centre+  number: '0.2886751345948128822545743902509787278238008756350634380093011632419888361514666728468576977928747626'+  comment: $\delta(A_{2})=\frac{\sqrt{3}}{6}$; $\det=3$, $\mu=2$, kissing number $6$;+    $A_{2}=\Lambda_{2}$+- params:+    family: A+    n: '2'+    expression: hermite+  number: '1.154700538379251529018297561003914911295203502540253752037204652967955344605866691387430791171499050'+  comment: $\gamma(A_{2})=\frac{2}{\sqrt{3}}$; Hermite's constant $\gamma_{2}$ CITE{Lagrange};+    $A_{2}=\Lambda_{2}$+  equals: HREF{Algebraic_numbers_of_degree_2#3,0,-4,2}+- params:+    family: A+    n: '3'+    expression: density+  number: '0.7404804896930610411693134983434489497691036148959483705142326011594057988499123184292211557941275396'+  comment: $\Delta(A_{3})=\frac{\sqrt{2}\,\pi}{6}$; the densest packing of any kind+    in dimension 3 CITE{Hales}; $A_{3}=\Lambda_{3}$; $A_3=D_3$, the face-centred cubic+    lattice+- params:+    family: A+    n: '3'+    expression: centre+  number: '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+- params:+    family: A+    n: '3'+    expression: hermite+  number: '1.259921049894873164767210607278228350570251464701507980081975112155299676513959483729396562436255094'+  comment: $\gamma(A_{3})=2^{1/3}$; Hermite's constant $\gamma_{3}$ CITE{Gauss}; $A_{3}=\Lambda_{3}$;+    $A_3=D_3$, the face-centred cubic lattice+- params:+    family: A+    n: '4'+    expression: density+  number: '0.5517276587966726327585962310645031466095675561351147374547792204419356284674867247526363800082606285'+  comment: $\Delta(A_{4})=\frac{\sqrt{5}\,\pi^{2}}{40}$+- params:+    family: A+    n: '4'+    expression: centre+  number: '0.1118033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137'+  comment: $\delta(A_{4})=\frac{\sqrt{5}}{20}$; $\det=5$, $\mu=2$, kissing number+    $20$+- params:+    family: A+    n: '4'+    expression: hermite+  number: '1.337480609952844048006466146517295872776070383304955129539966906297538647242720943449191470376352353'+  comment: $\gamma(A_{4})=\frac{2}{5^{1/4}}$+- params:+    family: A+    n: '5'+    expression: density+  number: '0.3798812505176037581053993966982966063494607803638771779523670617473220782275171526904027213139341401'+  comment: $\Delta(A_{5})=\frac{\sqrt{3}\,\pi^{2}}{45}$+- params:+    family: A+    n: '5'+    expression: centre+  number: '0.07216878364870322056364359756274468195595021890876585950232529081049720903786666821171442444821869065'+  comment: $\delta(A_{5})=\frac{\sqrt{3}}{24}$; $\det=6$, $\mu=2$, kissing number+    $30$+- params:+    family: A+    n: '5'+    expression: hermite+  number: '1.397654237543158490465954205542839687013584633139712792584031713336178783783983105452602535006347153'+  comment: $\gamma(A_{5})=\frac{2^{4/5}}{3^{1/5}}$+- params:+    family: A+    n: '6'+    expression: density+  number: '0.2441514796968294369713326019190372269072956097640023376940945400601858116393881678377464623771082860'+  comment: $\Delta(A_{6})=\frac{\sqrt{7}\,\pi^{3}}{336}$+- params:+    family: A+    n: '6'+    expression: centre+  number: '0.04724555912615340340181456702927250760196891398361518179229168677144765755768363621000701582990256448'+  comment: $\delta(A_{6})=\frac{\sqrt{7}}{56}$; $\det=7$, $\mu=2$, kissing number+    $42$+- params:+    family: A+    n: '6'+    expression: hermite+  number: '1.446040052798967552509685118554498177089689708920244836572156252251105069335891595768411780348917303'+  comment: $\gamma(A_{6})=\frac{2}{7^{1/6}}$+- params:+    family: A+    n: '7'+    expression: density+  number: '0.1476489365728562865498872146052447390582156598375481318768787528752685472260271716074604811076327914'+  comment: $\Delta(A_{7})=\frac{\pi^{3}}{210}$+- params:+    family: A+    n: '7'+    expression: centre+  number: 1/32+  comment: $\delta(A_{7})=\frac{1}{32}$; $\det=8$, $\mu=2$, kissing number $56$+- params:+    family: A+    n: '7'+    expression: hermite+  number: '1.485994289136948424799853286714592606323711359437109733356503197150790858340556200496397994091233511'+  comment: $\gamma(A_{7})=2^{4/7}$+- params:+    family: A+    n: '8'+    expression: density+  number: '0.08455650263368267121218778879227874240427741811865053966273251683157192313960694113725856530794530603'+  comment: $\Delta(A_{8})=\frac{\pi^{4}}{1152}$+- params:+    family: A+    n: '8'+    expression: centre+  number: 1/48+  comment: $\delta(A_{8})=\frac{1}{48}$; $\det=9$, $\mu=2$, kissing number $72$+- params:+    family: A+    n: '8'+    expression: hermite+  number: '1.519671371303185094662375501309090670793546897774620637222577307400644466342094543188821228330074314'+  comment: $\gamma(A_{8})=\frac{2}{3^{1/4}}$+- params:+    family: A+    n: '9'+    expression: density+  number: '0.04609806331819994245073740620365313147351893372623936529917451763099279502435109726330283538937166342'+  comment: $\Delta(A_{9})=\frac{\sqrt{5}\,\pi^{4}}{4725}$+- params:+    family: A+    n: '9'+    expression: centre+  number: '0.01397542485937368560255733542957047647150386474757203577669310778381575578523628062134009005236738922'+  comment: $\delta(A_{9})=\frac{\sqrt{5}}{160}$; $\det=10$, $\mu=2$, kissing number+    $90$+- params:+    family: A+    n: '9'+    expression: hermite+  number: '1.548527365362254119453358903073938556428667487479489470582783954818172283985407781713774776894218079'+  comment: $\gamma(A_{9})=\frac{2^{8/9}}{5^{1/9}}$+- params:+    family: A+    n: '10'+    expression: density+  number: '0.02402823089241500106101615809674920988862086845323238560585835100840754771083244875776808094231462279'+  comment: $\Delta(A_{10})=\frac{\sqrt{11}\,\pi^{5}}{42240}$+- params:+    family: A+    n: '10'+    expression: centre+  number: '0.009422229518055113207712877092814450806611047004515209082553074278740013189230238200877396020250825713'+  comment: $\delta(A_{10})=\frac{\sqrt{11}}{352}$; $\det=11$, $\mu=2$, kissing number+    $110$+- params:+    family: A+    n: '10'+    expression: hermite+  number: '1.573586884393544470910577389703060847050289960034329483222479690112842968652442417915296777326848230'+  comment: $\gamma(A_{10})=\frac{2}{11^{1/10}}$+- params:+    family: A+    n: '11'+    expression: density+  number: '0.01201847168464467383503297013093922603134950303996848600699281026471812948055207148382095348220641627'+  comment: $\Delta(A_{11})=\frac{\sqrt{6}\,\pi^{5}}{62370}$+- params:+    family: A+    n: '11'+    expression: centre+  number: '0.006378879538497859630722093944546592166577988230876745126126803560549375982961757881614217273710000612'+  comment: $\delta(A_{11})=\frac{\sqrt{6}}{384}$; $\det=12$, $\mu=2$, kissing number+    $132$+- params:+    family: A+    n: '11'+    expression: hermite+  number: '1.595594790350010671043327255349365001093923225034600716432474987935317816928421338613405013027077721'+  comment: $\gamma(A_{11})=\frac{2^{9/11}}{3^{1/11}}$+- params:+    family: A+    n: '12'+    expression: density+  number: '0.005786488436686590234019311923709057050834093105411363221367165329909345462276874389270809142187284009'+  comment: $\Delta(A_{12})=\frac{\sqrt{13}\,\pi^{6}}{599040}$+- params:+    family: A+    n: '12'+    expression: centre+  number: '0.004333595283009602515768294792632807627705885305102459390276986846426883351313714477409397935118378266'+  comment: $\delta(A_{12})=\frac{\sqrt{13}}{832}$; $\det=13$, $\mu=2$, kissing number+    $156$+- params:+    family: A+    n: '12'+    expression: hermite+  number: '1.615107310781479943692032140716542329654195974804270993044820930949102699395594766266221536198739415'+  comment: $\gamma(A_{12})=\frac{2}{13^{1/12}}$+- params:+    family: A+    n: '13'+    expression: density+  number: '0.002688947792255565796332472953030458916732744761264711054285497493156665660148928078704263168035576243'+  comment: $\Delta(A_{13})=\frac{\sqrt{7}\,\pi^{6}}{945945}$+- params:+    family: A+    n: '13'+    expression: centre+  number: '0.002952847445384587712613410439329531725123057123975948862018230423215478597355227263125438489368910280'+  comment: $\delta(A_{13})=\frac{\sqrt{7}}{896}$; $\det=14$, $\mu=2$, kissing number+    $182$+- params:+    family: A+    n: '13'+    expression: hermite+  number: '1.632549283341321630287408419015642589110934389637549360001422436946971075177029699991942659693631158'+  comment: $\gamma(A_{13})=\frac{2^{12/13}}{7^{1/13}}$+- params:+    family: A+    n: '14'+    expression: density+  number: '0.001208823719808465618429741353024457419416352973384621441670032941077273546225795968453029609363729344'+  comment: $\Delta(A_{14})=\frac{\sqrt{15}\,\pi^{7}}{9676800}$+- params:+    family: A+    n: '14'+    expression: centre+  number: '0.002017178826149696294364200729053333130642146721506036888847694669850772443717176579957962175447225791'+  comment: $\delta(A_{14})=\frac{\sqrt{15}}{1920}$; $\det=15$, $\mu=2$, kissing number+    $210$+- params:+    family: A+    n: '14'+    expression: hermite+  number: '1.648251490557794949911981416796352816339300520838035539580595781886398192656062395453211248896066979'+  comment: $\gamma(A_{14})=\frac{2}{3^{1/14}\cdot 5^{1/14}}$+- params:+    family: A+    n: '15'+    expression: density+  number: '0.0005267990830238343014942344870440350007947798198889932986102095778021866547206442444659482856140843291'+  comment: $\Delta(A_{15})=\frac{\sqrt{2}\,\pi^{7}}{8108100}$+- params:+    family: A+    n: '15'+    expression: centre+  number: '0.001381067932004975633595399144736033279853195190797800852711601306631574685998151405127331576491837473'+  comment: $\delta(A_{15})=\frac{\sqrt{2}}{1024}$; $\det=16$, $\mu=2$, kissing number+    $240$+- params:+    family: A+    n: '15'+    expression: hermite+  number: '1.662475792285575556085280683775664373100417273320547235153185782890535235340516306696443430280054256'+  comment: $\gamma(A_{15})=2^{11/15}$+- params:+    family: A+    n: '16'+    expression: density+  number: '0.0002229533653292515595032608571780731988836775933466674830516920285674050900256010281864321426646726454'+  comment: $\Delta(A_{16})=\frac{\sqrt{17}\,\pi^{8}}{175472640}$+- params:+    family: A+    n: '16'+    expression: centre+  number: '0.0009474047852981756778082283676411022576165439396538649895217448467589509067871373147030936697634844432'+  comment: $\delta(A_{16})=\frac{\sqrt{17}}{4352}$; $\det=17$, $\mu=2$, kissing number+    $272$+- params:+    family: A+    n: '16'+    expression: hermite+  number: '1.675432509473520540573916523910756553754036977519787999855545909138671812239108332465310135146355849'+  comment: $\gamma(A_{16})=\frac{2}{17^{1/16}}$+- params:+    family: A+    n: '17'+    expression: density+  number: '0.00009178457481584572781010880191129873077004989173635449703634628663582274807575746737924834136590995668'+  comment: $\Delta(A_{17})=\frac{\pi^{8}}{103378275}$+- params:+    family: A+    n: '17'+    expression: centre+  number: 1/1536+  comment: $\delta(A_{17})=\frac{1}{1536}$; $\det=18$, $\mu=2$, kissing number $306$+- params:+    family: A+    n: '17'+    expression: hermite+  number: '1.687292734883595117432729236880446158088124119992776131174272823042824383509922794626688643876976786'+  comment: $\gamma(A_{17})=\frac{2^{16/17}}{3^{2/17}}$+- params:+    family: A+    n: '18'+    expression: density+  number: '0.00003680773215105456382239482196898414195518801818437606034613107298917086428366134564807742509518674705'+  comment: $\Delta(A_{18})=\frac{\sqrt{19}\,\pi^{9}}{3530096640}$+- params:+    family: A+    n: '18'+    expression: centre+  number: '0.0004480776052159409490375187072224111491711558311299799482823133365706781792971996387393255711196489028'+  comment: $\delta(A_{18})=\frac{\sqrt{19}}{9728}$; $\det=19$, $\mu=2$, kissing number+    $342$+- params:+    family: A+    n: '18'+    expression: hermite+  number: '1.698197233107663644258114840959473123346329995662803593305588173564765286009819641957041149154603362'+  comment: $\gamma(A_{18})=\frac{2}{19^{1/18}}$+- params:+    family: A+    n: '19'+    expression: density+  number: '0.00001439750463073555763471912245959366331864715084979751906909650899098828277887365377359187405187304647'+  comment: $\Delta(A_{19})=\frac{\sqrt{10}\,\pi^{9}}{6547290750}$+- params:+    family: A+    n: '19'+    expression: centre+  number: '0.0003088161777508182941405169476985076693085503065747282057478032082805268006483631075531492293339160445'+  comment: $\delta(A_{19})=\frac{\sqrt{10}}{10240}$; $\det=20$, $\mu=2$, kissing number+    $380$+- params:+    family: A+    n: '19'+    expression: hermite+  number: '1.708262993375513107747813508738544085881574715989018696495543082141575575582458905876660260792716515'+  comment: $\gamma(A_{19})=\frac{2^{17/19}}{5^{1/19}}$+- params:+    family: A+    n: '20'+    expression: density+  number: '0.000005499536516287368162585488027923258154810059003100507260256746508041106132330487791988321874212521457'+  comment: $\Delta(A_{20})=\frac{\sqrt{21}\,\pi^{10}}{78033715200}$+- params:+    family: A+    n: '20'+    expression: centre+  number: '0.0002131034084335863098301733256011908709535182559880939314828516612679905331820950003071224683544198769'+  comment: $\delta(A_{20})=\frac{\sqrt{21}}{21504}$; $\det=21$, $\mu=2$, kissing number+    $420$+- params:+    family: A+    n: '20'+    expression: hermite+  number: '1.717588133299478501121941496385791503870438068950841220941884937760204197578598837351277574516192101'+  comment: $\gamma(A_{20})=\frac{2}{3^{1/20}\cdot 7^{1/20}}$+- params:+    family: A+    n: '21'+    expression: density+  number: '0.000002053626511494816144477693541305358043241149208237410748849077115300199226698615359870862561598339641'+  comment: $\Delta(A_{21})=\frac{\sqrt{11}\,\pi^{10}}{151242416325}$+- params:+    family: A+    n: '21'+    expression: centre+  number: '0.0001472223362196111438705137045752257938532976094455501419148917856053127060817224718887093128164191518'+  comment: $\delta(A_{21})=\frac{\sqrt{11}}{22528}$; $\det=22$, $\mu=2$, kissing number+    $462$+- params:+    family: A+    n: '21'+    expression: hermite+  number: '1.726255622473788359715165126997843588765328008490425107640098302638643331340785087011837687400754226'+  comment: $\gamma(A_{21})=\frac{2^{20/21}}{11^{1/21}}$+- params:+    family: A+    n: '22'+    expression: density+  number: '7.504106884735152640492964258639967853758404908248139824753027760630275003076909705640907794684904389e-7'+  comment: $\Delta(A_{22})=\frac{\sqrt{23}\,\pi^{11}}{1880240947200}$+- params:+    family: A+    n: '22'+    expression: centre+  number: '0.0001018136787388060364639401762942147995923213961002065503245012974364864392813235411534517545524142081'+  comment: $\delta(A_{22})=\frac{\sqrt{23}}{47104}$; $\det=23$, $\mu=2$, kissing number+    $506$+- params:+    family: A+    n: '22'+    expression: hermite+  number: '1.734336147394595049711209879648012028284610026953493822420039353140156496077575847807467040691158054'+  comment: $\gamma(A_{22})=\frac{2}{23^{1/22}}$+- params:+    family: A+    n: '23'+    expression: density+  number: '2.685648792405079381122957708162599715143941715032483899447864135464203324732931460809555213349874405e-7'+  comment: $\Delta(A_{23})=\frac{\sqrt{3}\,\pi^{11}}{1897404859350}$+- params:+    family: A+    n: '23'+    expression: centre+  number: '0.00007047732778193673883168320074486785347260763565309165967023954180712618070104166817550236762521356509'+  comment: $\delta(A_{23})=\frac{\sqrt{3}}{24576}$; $\det=24$, $\mu=2$, kissing number+    $552$+- params:+    family: A+    n: '23'+    expression: hermite+  number: '1.741890342175930340199256595687853411363891921062025217911179240798942327455776880768285594628532936'+  comment: $\gamma(A_{23})=\frac{2^{20/23}}{3^{1/23}}$+- params:+    family: A+    n: '24'+    expression: density+  number: '9.421749557636343007340554510185340039885116284181168712083240319001443526616140290465564932549530827e-8'+  comment: $\Delta(A_{24})=\frac{\pi^{12}}{9809952768000}$+- params:+    family: A+    n: '24'+    expression: centre+  number: 1/20480+  comment: $\delta(A_{24})=\frac{1}{20480}$; $\det=25$, $\mu=2$, kissing number $600$+- params:+    family: A+    n: '24'+    expression: hermite+  number: '1.748970544442335671271718425882761979326529471699088254127749773872966802758875884305067209164683923'+  comment: $\gamma(A_{24})=\frac{2}{5^{1/12}}$+- params:+    family: D+    n: '4'+    expression: density+  number: '0.6168502750680849136771556874922594459571062129525494141508343360137528014012003276876108377324095145'+  comment: $\Delta(D_{4})=\frac{\pi^{2}}{16}$; the densest lattice packing in dimension+    4 CITE{KZ}; $D_{4}=\Lambda_{4}$+- params:+    family: D+    n: '4'+    expression: centre+  number: 1/8+  comment: $\delta(D_{4})=\frac{1}{8}$; $\det=4$, $\mu=2$, kissing number $24$; $D_{4}=\Lambda_{4}$+- params:+    family: D+    n: '4'+    expression: hermite+  number: '1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573'+  comment: $\gamma(D_{4})=\sqrt{2}$; Hermite's constant $\gamma_{4}$ CITE{KZ}; $D_{4}=\Lambda_{4}$+  equals: HREF{Algebraic_numbers_of_degree_2#1,0,-2,2}+- params:+    family: D+    n: '5'+    expression: density+  number: '0.4652576133092586356105040624112936859946577513965361577435664445013271841888718143111600891540540958'+  comment: $\Delta(D_{5})=\frac{\sqrt{2}\,\pi^{2}}{30}$; the densest lattice packing+    in dimension 5 CITE{KZ}; $D_{5}=\Lambda_{5}$+- params:+    family: D+    n: '5'+    expression: centre+  number: '0.08838834764831844055010554526310612991060449221105925457354248362442077990388168992814922089547759830'+  comment: $\delta(D_{5})=\frac{\sqrt{2}}{16}$; $\det=4$, $\mu=2$, kissing number+    $40$; $D_{5}=\Lambda_{5}$+- params:+    family: D+    n: '5'+    expression: hermite+  number: '1.515716566510398082347259801306445238681283542978141642037505242097453677202058277641176134849431791'+  comment: $\gamma(D_{5})=2^{3/5}$; Hermite's constant $\gamma_{5}$ CITE{KZ}; $D_{5}=\Lambda_{5}$+- params:+    family: D+    n: '6'+    expression: density+  number: '0.3229820487531231268278782819489728666898467558946365384806722719146499470569344378913198024229467312'+  comment: $\Delta(D_{6})=\frac{\pi^{3}}{96}$+- params:+    family: D+    n: '6'+    expression: centre+  number: 1/16+  comment: $\delta(D_{6})=\frac{1}{16}$; $\det=4$, $\mu=2$, kissing number $60$+- params:+    family: D+    n: '6'+    expression: hermite+  number: '1.587401051968199474751705639272308260391493327899853009808285761825216505624219173273544213262220957'+  comment: $\gamma(D_{6})=2^{2/3}$+- params:+    family: D+    n: '7'+    expression: density+  number: '0.2088071285712982487119759477845852987347505409091042469244108708947825315491728189023710095441813824'+  comment: $\Delta(D_{7})=\frac{\sqrt{2}\,\pi^{3}}{210}$+- params:+    family: D+    n: '7'+    expression: centre+  number: '0.04419417382415922027505277263155306495530224610552962728677124181221038995194084496407461044773879915'+  comment: $\delta(D_{7})=\frac{\sqrt{2}}{32}$; $\det=4$, $\mu=2$, kissing number+    $84$+- params:+    family: D+    n: '7'+    expression: hermite+  number: '1.640670712015275862340569365734992099789404915306025246598383626990030803453104404313461505028108899'+  comment: $\gamma(D_{7})=2^{5/7}$+- params:+    family: D+    n: '8'+    expression: density+  number: '0.1268347539505240068182816831884181136064161271779758094940987752473578847094104117058878479619179591'+  comment: $\Delta(D_{8})=\frac{\pi^{4}}{768}$+- params:+    family: D+    n: '8'+    expression: centre+  number: 1/32+  comment: $\delta(D_{8})=\frac{1}{32}$; $\det=4$, $\mu=2$, kissing number $112$+- params:+    family: D+    n: '8'+    expression: hermite+  number: '1.681792830507429086062250952466429790080068524713569021626452171949849509907804479628648008398585072'+  comment: $\gamma(D_{8})=2^{3/4}$+- params:+    family: D+    n: '9'+    expression: density+  number: '0.07288743790408555526660687722882599977128699625536283354816057423198158865453764503620500157335241281'+  comment: $\Delta(D_{9})=\frac{\sqrt{2}\,\pi^{4}}{1890}$+- params:+    family: D+    n: '9'+    expression: centre+  number: '0.02209708691207961013752638631577653247765112305276481364338562090610519497597042248203730522386939957'+  comment: $\delta(D_{9})=\frac{\sqrt{2}}{64}$; $\det=4$, $\mu=2$, kissing number+    $144$+- params:+    family: D+    n: '9'+    expression: hermite+  number: '1.714487965706145661766070110832072051511440720277524415717680108293178339505411178988245367996736297'+  comment: $\gamma(D_{9})=2^{7/9}$+- params:+    family: D+    n: '10'+    expression: density+  number: '0.03984631312308352256025277474523901126045582117553059772831932860497851460073099277158639329082097902'+  comment: $\Delta(D_{10})=\frac{\pi^{5}}{7680}$+- params:+    family: D+    n: '10'+    expression: centre+  number: 1/64+  comment: $\delta(D_{10})=\frac{1}{64}$; $\det=4$, $\mu=2$, kissing number $180$+- params:+    family: D+    n: '10'+    expression: hermite+  number: '1.741101126592248278272540034959492197958250848696006096483719137013500035504956020376757529743340835'+  comment: $\gamma(D_{10})=2^{4/5}$+- params:+    family: D+    n: '11'+    expression: density+  number: '0.02081660358713249246851755142559941854696573947086251038321947354173607770740169453703408263008532520'+  comment: $\Delta(D_{11})=\frac{\sqrt{2}\,\pi^{5}}{20790}$+- params:+    family: D+    n: '11'+    expression: centre+  number: '0.01104854345603980506876319315788826623882556152638240682169281045305259748798521124101865261193469979'+  comment: $\delta(D_{11})=\frac{\sqrt{2}}{128}$; $\det=4$, $\mu=2$, kissing number+    $220$+- params:+    family: D+    n: '11'+    expression: hermite+  number: '1.763182509992042383391871761763604832617070787458418968156890322075515373842549813095564618098758414'+  comment: $\gamma(D_{11})=2^{9/11}$+- params:+    family: D+    n: '12'+    expression: density+  number: '0.01043174038167648043652581861602018119430574574740718984311063571488817179733452922633703785972130790'+  comment: $\Delta(D_{12})=\frac{\pi^{6}}{92160}$+- params:+    family: D+    n: '12'+    expression: centre+  number: 1/128+  comment: $\delta(D_{12})=\frac{1}{128}$; $\det=4$, $\mu=2$, kissing number $264$+- params:+    family: D+    n: '12'+    expression: hermite+  number: '1.781797436280678609480452411181025015974425231756320806767513984503861606631524985275051534501114395'+  comment: $\gamma(D_{12})=2^{5/6}$+- params:+    family: D+    n: '13'+    expression: density+  number: '0.005030560684771259670778501933859200113077304220606558325382183414811115955907235458098968860901921006'+  comment: $\Delta(D_{13})=\frac{\sqrt{2}\,\pi^{6}}{270270}$+- params:+    family: D+    n: '13'+    expression: centre+  number: '0.005524271728019902534381596578944133119412780763191203410846405226526298743992605620509326305967349893'+  comment: $\delta(D_{13})=\frac{\sqrt{2}}{256}$; $\det=4$, $\mu=2$, kissing number+    $312$+- params:+    family: D+    n: '13'+    expression: hermite+  number: '1.797701946391083748217080870549112685362805768801854913219969080072771197585775002464001567572728231'+  comment: $\gamma(D_{13})=2^{11/13}$+- params:+    family: D+    n: '14'+    expression: density+  number: '0.002340877067659344050342731966976718929747514009250528787476930255133257895657871311491236222012966375'+  comment: $\Delta(D_{14})=\frac{\pi^{7}}{1290240}$+- params:+    family: D+    n: '14'+    expression: centre+  number: 1/256+  comment: $\delta(D_{14})=\frac{1}{256}$; $\det=4$, $\mu=2$, kissing number $364$+- params:+    family: D+    n: '14'+    expression: hermite+  number: '1.811447328527813343188345746430206375400891762515874710237416262768844934627125673909528787782071557'+  comment: $\gamma(D_{14})=2^{6/7}$+- params:+    family: D+    n: '15'+    expression: density+  number: '0.001053598166047668602988468974088070001589559639777986597220419155604373309441288488931896571228168658'+  comment: $\Delta(D_{15})=\frac{\sqrt{2}\,\pi^{7}}{4054050}$+- params:+    family: D+    n: '15'+    expression: centre+  number: '0.002762135864009951267190798289472066559706390381595601705423202613263149371996302810254663152983674947'+  comment: $\delta(D_{15})=\frac{\sqrt{2}}{512}$; $\det=4$, $\mu=2$, kissing number+    $420$+- params:+    family: D+    n: '15'+    expression: hermite+  number: '1.823444977116433615632210157088315446532424756712371891434363603649620034805593810642955674651026071'+  comment: $\gamma(D_{15})=2^{13/15}$+- params:+    family: D+    n: '16'+    expression: density+  number: '0.0004596301374197132901208581108895809014552127642745285127238968823177717373236460026833895402705560171'+  comment: $\Delta(D_{16})=\frac{\pi^{8}}{20643840}$+- params:+    family: D+    n: '16'+    expression: centre+  number: 1/512+  comment: $\delta(D_{16})=\frac{1}{512}$; $\det=4$, $\mu=2$, kissing number $480$+- params:+    family: D+    n: '16'+    expression: hermite+  number: '1.834008086409342463487083189588288856077310332873679499583324138706477662244694725682930978761320627'+  comment: $\gamma(D_{16})=2^{7/8}$+- params:+    family: D+    n: '17'+    expression: density+  number: '0.0001947044857818255767834649600340239148692965034703838084254511881296855370101989179443183260286988034'+  comment: $\Delta(D_{17})=\frac{\sqrt{2}\,\pi^{8}}{68918850}$+- params:+    family: D+    n: '17'+    expression: centre+  number: '0.001381067932004975633595399144736033279853195190797800852711601306631574685998151405127331576491837473'+  comment: $\delta(D_{17})=\frac{\sqrt{2}}{1024}$; $\det=4$, $\mu=2$, kissing number+    $544$+- params:+    family: D+    n: '17'+    expression: hermite+  number: '1.843379281881730798286822083933046561318432049217838373392893084752710250315544089737509611615431545'+  comment: $\gamma(D_{17})=2^{15/17}$+- params:+    family: D+    n: '18'+    expression: density+  number: '0.00008022059239367991093633043508173666485140377031030578429387587028343190143434003721509597996456690973'+  comment: $\Delta(D_{18})=\frac{\pi^{9}}{371589120}$+- params:+    family: D+    n: '18'+    expression: centre+  number: 1/1024+  comment: $\delta(D_{18})=\frac{1}{1024}$; $\det=4$, $\mu=2$, kissing number $612$+- params:+    family: D+    n: '18'+    expression: hermite+  number: '1.851749424574580858409381921539531664895374719178026270189414539251873052022230063371359777734573426'+  comment: $\gamma(D_{18})=2^{8/9}$+- params:+    family: D+    n: '19'+    expression: density+  number: '0.00003219379906069271485067828951841922287994012647824365295830751909125131408072314178950984246778347148'+  comment: $\Delta(D_{19})=\frac{\sqrt{2}\,\pi^{9}}{1309458150}$+- params:+    family: D+    n: '19'+    expression: centre+  number: '0.0006905339660024878167976995723680166399265975953989004263558006533157873429990757025636657882459187367'+  comment: $\delta(D_{19})=\frac{\sqrt{2}}{2048}$; $\det=4$, $\mu=2$, kissing number+    $684$+- params:+    family: D+    n: '19'+    expression: hermite+  number: '1.859270710016812562123254411861692017391741056754022418807623253650390951186565003295635179484872202'+  comment: $\gamma(D_{19})=2^{17/19}$+- params:+    family: D+    n: '20'+    expression: density+  number: '0.00001260102118653030274052651086567326643417306691804037401943475220603852018356824016384215735021313726'+  comment: $\Delta(D_{20})=\frac{\pi^{10}}{7431782400}$+- params:+    family: D+    n: '20'+    expression: centre+  number: 1/2048+  comment: $\delta(D_{20})=\frac{1}{2048}$; $\det=4$, $\mu=2$, kissing number $760$+- params:+    family: D+    n: '20'+    expression: hermite+  number: '1.866065983073614831962686532299884334054459928702988077800994770970794040621127095207414946789290627'+  comment: $\gamma(D_{20})=2^{9/10}$+- params:+    family: D+    n: '21'+    expression: density+  number: '0.000004816181077153248527235069969072587297983772242769206991389298988054133465637759707629929029225514467'+  comment: $\Delta(D_{21})=\frac{\sqrt{2}\,\pi^{10}}{27498621150}$+- params:+    family: D+    n: '21'+    expression: centre+  number: '0.0003452669830012439083988497861840083199632987976994502131779003266578936714995378512818328941229593683'+  comment: $\delta(D_{21})=\frac{\sqrt{2}}{4096}$; $\det=4$, $\mu=2$, kissing number+    $840$+- params:+    family: D+    n: '21'+    expression: hermite+  number: '1.872235484907209776378711215229745504726876897943956214601289522670065829941920477891890268718690680'+  comment: $\gamma(D_{21})=2^{19/21}$+- params:+    family: D+    n: '22'+    expression: density+  number: '0.000001799421617606042670229270009104194702444247616216063830541953787553098187456566884103255944885519082'+  comment: $\Delta(D_{22})=\frac{\pi^{11}}{163499212800}$+- params:+    family: D+    n: '22'+    expression: centre+  number: 1/4096+  comment: $\delta(D_{22})=\frac{1}{4096}$; $\det=4$, $\mu=2$, kissing number $924$+- params:+    family: D+    n: '22'+    expression: hermite+  number: '1.877861821323412700576744871323752854765976936737945629652613867420387780642365977698176175232641815'+  comment: $\gamma(D_{22})=2^{10/11}$+- params:+    family: D+    n: '23'+    expression: density+  number: '6.578469169714270766418035999085643475314254400414369966793037041392389567346920524014085263231902891e-7'+  comment: $\Delta(D_{23})=\frac{\sqrt{2}\,\pi^{11}}{632468286450}$+- params:+    family: D+    n: '23'+    expression: centre+  number: '0.0001726334915006219541994248930920041599816493988497251065889501633289468357497689256409164470614796842'+  comment: $\delta(D_{23})=\frac{\sqrt{2}}{8192}$; $\det=4$, $\mu=2$, kissing number+    $1012$+- params:+    family: D+    n: '23'+    expression: hermite+  number: '1.883013676297781883635519373298952450378415489447049402291517808983205147730849390317787227894124354'+  comment: $\gamma(D_{23})=2^{21/23}$+- params:+    family: D+    n: '24'+    expression: density+  number: '2.355437389409085751835138627546335009971279071045292178020810079750360881654035072616391233137382707e-7'+  comment: $\Delta(D_{24})=\frac{\pi^{12}}{3923981107200}$+- params:+    family: D+    n: '24'+    expression: centre+  number: 1/8192+  comment: $\delta(D_{24})=\frac{1}{8192}$; $\det=4$, $\mu=2$, kissing number $1104$+- params:+    family: D+    n: '24'+    expression: hermite+  number: '1.887748625363386993283826313335068752015136606677485627484250284636572975477413406090396909221235416'+  comment: $\gamma(D_{24})=2^{11/12}$+- params:+    family: E+    n: '6'+    expression: density+  number: '0.3729475455820649395634775586799581063936647972683873631114040655972831720296832195225267216353405428'+  comment: $\Delta(E_{6})=\frac{\sqrt{3}\,\pi^{3}}{144}$; the densest lattice packing+    in dimension 6 CITE{Blichfeldt}; $E_{6}=\Lambda_{6}$+- params:+    family: E+    n: '6'+    expression: centre+  number: '0.07216878364870322056364359756274468195595021890876585950232529081049720903786666821171442444821869065'+  comment: $\delta(E_{6})=\frac{\sqrt{3}}{24}$; $\det=3$, $\mu=2$, kissing number+    $72$; $E_{6}=\Lambda_{6}$+- params:+    family: E+    n: '6'+    expression: hermite+  number: '1.665366355311208639217572725017671513324124095787337672980480482451078485985633426184051324797373192'+  comment: $\gamma(E_{6})=\frac{2}{3^{1/6}}$; Hermite's constant $\gamma_{6}$ CITE{Blichfeldt};+    $E_{6}=\Lambda_{6}$+- params:+    family: E+    n: '7'+    expression: density+  number: '0.2952978731457125730997744292104894781164313196750962637537575057505370944520543432149209622152655828'+  comment: $\Delta(E_{7})=\frac{\pi^{3}}{105}$; the densest lattice packing in dimension+    7 CITE{Blichfeldt}; $E_{7}=\Lambda_{7}$+- params:+    family: E+    n: '7'+    expression: centre+  number: 1/16+  comment: $\delta(E_{7})=\frac{1}{16}$; $\det=2$, $\mu=2$, kissing number $126$;+    $E_{7}=\Lambda_{7}$+- params:+    family: E+    n: '7'+    expression: hermite+  number: '1.811447328527813343188345746430206375400891762515874710237416262768844934627125673909528787782071557'+  comment: $\gamma(E_{7})=2^{6/7}$; Hermite's constant $\gamma_{7}$ CITE{Blichfeldt};+    $E_{7}=\Lambda_{7}$+- params:+    family: E+    n: '8'+    expression: density+  number: '0.2536695079010480136365633663768362272128322543559516189881975504947157694188208234117756959238359181'+  comment: $\Delta(E_{8})=\frac{\pi^{4}}{384}$; the densest packing of any kind in+    dimension 8 CITE{Viazovska}; $E_{8}=\Lambda_{8}$+- params:+    family: E+    n: '8'+    expression: centre+  number: 1/16+  comment: $\delta(E_{8})=\frac{1}{16}$; $\det=1$, $\mu=2$, kissing number $240$;+    $E_{8}=\Lambda_{8}$+- params:+    family: E+    n: '8'+    expression: hermite+  number: '2'+  comment: $\gamma(E_{8})=2$; Hermite's constant $\gamma_{8}$ CITE{Blichfeldt}; $E_{8}=\Lambda_{8}$+- params:+    family: A*+    n: '3'+    expression: density+  number: '0.6801747615878316939727793466158083996065248430347777158387068507705461929223827063815468705795225633'+  comment: $\Delta(A_{3}^{*})=\frac{\sqrt{3}\,\pi}{8}$; $A_3^{*}$ is the body-centred+    cubic lattice+- params:+    family: A*+    n: '3'+    expression: centre+  number: '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+- params:+    family: A*+    n: '3'+    expression: hermite+  number: '1.190550788976149606063779229454231195293619995924889757356214321368912379218164379955158159946665718'+  comment: $\gamma(A_{3}^{*})=\frac{3}{2^{4/3}}$; $A_3^{*}$ is the body-centred cubic+    lattice+- params:+    family: A*+    n: '4'+    expression: density+  number: '0.4413821270373381062068769848516025172876540449080917899638233763535485027739893798021091040066085028'+  comment: $\Delta(A_{4}^{*})=\frac{\sqrt{5}\,\pi^{2}}{50}$+- params:+    family: A*+    n: '4'+    expression: centre+  number: '0.08944271909999158785636694674925104941762473438446102897083588981642083702551219597657657633515129100'+  comment: $\delta(A_{4}^{*})=\frac{\sqrt{5}}{25}$; $\det=\frac{1}{5}$, $\mu=\frac{4}{5}$,+    kissing number $10$+- params:+    family: A*+    n: '4'+    expression: hermite+  number: '1.196279024976976433529519195312730716290767806091765076413274800038337173578397610278899540565990780'+  comment: $\gamma(A_{4}^{*})=\frac{4}{5^{3/4}}$+- params:+    family: A*+    n: '5'+    expression: density+  number: '0.2554294716651262188697204773446773826896146093218123784512866761305257539201327429410353611149354761'+  comment: $\Delta(A_{5}^{*})=\frac{5\sqrt{5}\,\pi^{2}}{432}$+- params:+    family: A*+    n: '5'+    expression: centre+  number: '0.04852578076171418611999074801934193219272175259573623533573995758269359647651486326854197934849787923'+  comment: $\delta(A_{5}^{*})=\frac{25\sqrt{5}}{1152}$; $\det=\frac{1}{6}$, $\mu=\frac{5}{6}$,+    kissing number $12$+- params:+    family: A*+    n: '5'+    expression: hermite+  number: '1.192474234254379584204353677619264087477208283051068116656973816465235418986027916546740133723444200'+  comment: $\gamma(A_{5}^{*})=\frac{5}{2^{4/5}\cdot 3^{4/5}}$+- params:+    family: A*+    n: '6'+    expression: density+  number: '0.1345324479962121387393057194247756148264690094617972064845010730943881002910914394207990711057535453'+  comment: $\Delta(A_{6}^{*})=\frac{9\sqrt{7}\,\pi^{3}}{5488}$+- params:+    family: A*+    n: '6'+    expression: centre+  number: '0.02603326727359473248671414917939505520924817709301244711003827638426707661341751383000386586545651512'+  comment: $\delta(A_{6}^{*})=\frac{27\sqrt{7}}{2744}$; $\det=\frac{1}{7}$, $\mu=\frac{6}{7}$,+    kissing number $14$+- params:+    family: A*+    n: '6'+    expression: hermite+  number: '1.185503617944418707977707297315562220751750916751643841759215107891221793818492532732982695834295264'+  comment: $\gamma(A_{6}^{*})=\frac{6}{7^{5/6}}$+- params:+    family: A*+    n: '7'+    expression: density+  number: '0.06542496682500976318841178317049200689781437042894750142896439628175291671274229810027112234011573601'+  comment: $\Delta(A_{7}^{*})=\frac{49\sqrt{7}\,\pi^{3}}{61440}$+- params:+    family: A*+    n: '7'+    expression: centre+  number: '0.01384723968040702167575155950162149545316496123958252581583158446511789865673808722414878479096631559'+  comment: $\delta(A_{7}^{*})=\frac{343\sqrt{7}}{65536}$; $\det=\frac{1}{8}$, $\mu=\frac{7}{8}$,+    kissing number $16$+- params:+    family: A*+    n: '7'+    expression: hermite+  number: '1.177662668553311615449978528282074166320466148430292132843997740402437682256855158817239670964254833'+  comment: $\gamma(A_{7}^{*})=\frac{7}{2^{18/7}}$+- params:+    family: A*+    n: '8'+    expression: density+  number: '0.02969336718000379126244180237424328951371058853000622517648768766650536669923096972721288438797530637'+  comment: $\Delta(A_{8}^{*})=\frac{2\pi^{4}}{6561}$+- params:+    family: A*+    n: '8'+    expression: centre+  number: 16/2187+  comment: $\delta(A_{8}^{*})=\frac{16}{2187}$; $\det=\frac{1}{9}$, $\mu=\frac{8}{9}$,+    kissing number $18$+- params:+    family: A*+    n: '8'+    expression: hermite+  number: '1.169843567068882187394861246041776937920060969071841934872820140974103817953157451600660577996979495'+  comment: $\gamma(A_{8}^{*})=\frac{8}{3^{7/4}}$+- params:+    family: A*+    n: '9'+    expression: density+  number: '0.01268057631496210299024375045179750466090203749203351293090715526701584886214768321832122164538123778'+  comment: $\Delta(A_{9}^{*})=\frac{729\pi^{4}}{5600000}$+- params:+    family: A*+    n: '9'+    expression: centre+  number: 19683/5120000+  comment: $\delta(A_{9}^{*})=\frac{19683}{5120000}$; $\det=\frac{1}{10}$, $\mu=\frac{9}{10}$,+    kissing number $20$+- params:+    family: A*+    n: '9'+    expression: hermite+  number: '1.162394698513395487869067991824878788678359736844704259274948851271907040790019706590131801511516636'+  comment: $\gamma(A_{9}^{*})=\frac{9}{2^{8/9}\cdot 5^{8/9}}$+- params:+    family: A*+    n: '10'+    expression: density+  number: '0.005128626565043158139175978010541717157430517991691223619855702950705114855293450062702360012617525867'+  comment: $\Delta(A_{10}^{*})=\frac{625\sqrt{11}\,\pi^{5}}{123687168}$+- params:+    family: A*+    n: '10'+    expression: centre+  number: '0.002011096731365496125544890438839229476856739422792843957583386184076397870114370219093085346853618629'+  comment: $\delta(A_{10}^{*})=\frac{3125\sqrt{11}}{5153632}$; $\det=\frac{1}{11}$,+    $\mu=\frac{10}{11}$, kissing number $22$+- params:+    family: A*+    n: '10'+    expression: hermite+  number: '1.155437832009218762368668306844038567203526505795372345829318449079678710773745742250058319348359151'+  comment: $\gamma(A_{10}^{*})=\frac{10}{11^{9/10}}$+- params:+    family: A*+    n: '11'+    expression: density+  number: '0.001974824150343715469595965443958393041762749590345920360474788772257660917585895503098313902280068258'+  comment: $\Delta(A_{11}^{*})=\frac{14641\sqrt{11}\,\pi^{5}}{7524679680}$+- params:+    family: A*+    n: '11'+    expression: centre+  number: '0.001048150355161516756873678341173930935669615508805010026688237873741051725255764306928505011101338770'+  comment: $\delta(A_{11}^{*})=\frac{161051\sqrt{11}}{509607936}$; $\det=\frac{1}{12}$,+    $\mu=\frac{11}{12}$, kissing number $24$+- params:+    family: A*+    n: '11'+    expression: hermite+  number: '1.148996815746165846296960885264741790700002400609945803764768929329747210685100065506658830954360798'+  comment: $\gamma(A_{11}^{*})=\frac{11}{2^{20/11}\cdot 3^{10/11}}$+- params:+    family: A*+    n: '12'+    expression: density+  number: '0.0007271195646081384619650923047635419083142301307217549548634271737744865157382171263821803032588546585'+  comment: $\Delta(A_{12}^{*})=\frac{81\sqrt{13}\,\pi^{6}}{386144720}$+- params:+    family: A*+    n: '12'+    expression: centre+  number: '0.0005445516654612287734368424986333603722080561303198830716247359856148450674774171952016678743226590763'+  comment: $\delta(A_{12}^{*})=\frac{729\sqrt{13}}{4826809}$; $\det=\frac{1}{13}$,+    $\mu=\frac{12}{13}$, kissing number $26$+- params:+    family: A*+    n: '12'+    expression: hermite+  number: '1.143053364832193642080725451708573220727260945388832966869381431266957926248229779523958599679964714'+  comment: $\gamma(A_{12}^{*})=\frac{12}{13^{11/12}}$+- params:+    family: A*+    n: '13'+    expression: density+  number: '0.0002569301036717477422742118134064993713459522224358610766287012328264141886487130659844282293808936008'+  comment: $\Delta(A_{13}^{*})=\frac{371293\sqrt{13}\,\pi^{6}}{5009249710080}$+- params:+    family: A*+    n: '13'+    expression: centre+  number: '0.0002821458276187352533077005154439994752258310794600173775156566999963893019384403148510638245706853397'+  comment: $\delta(A_{13}^{*})=\frac{4826809\sqrt{13}}{61681958912}$; $\det=\frac{1}{14}$,+    $\mu=\frac{13}{14}$, kissing number $28$+- params:+    family: A*+    n: '13'+    expression: hermite+  number: '1.137572308593258625637631643430085515600377313572473565145920520158833345042112686021445702022483204'+  comment: $\gamma(A_{13}^{*})=\frac{13}{2^{12/13}\cdot 7^{12/13}}$+- params:+    family: A*+    n: '14'+    expression: density+  number: '0.00008739804116826102174362552389301032529456474747962392721534278745842314079160982731392819652865701006'+  comment: $\Delta(A_{14}^{*})=\frac{117649\sqrt{15}\,\pi^{7}}{15746400000000}$+- params:+    family: A*+    n: '14'+    expression: centre+  number: '0.0001458421730171785424723908443133514733746765816159557651623415757689252951981278246091254908184875430'+  comment: $\delta(A_{14}^{*})=\frac{823543\sqrt{15}}{21870000000}$; $\det=\frac{1}{15}$,+    $\mu=\frac{14}{15}$, kissing number $30$+- params:+    family: A*+    n: '14'+    expression: hermite+  number: '1.132513258662339501526221675077404393975218377385689528415972103227012265105474198628376870811977194'+  comment: $\gamma(A_{14}^{*})=\frac{14}{3^{13/14}\cdot 5^{13/14}}$+- params:+    family: A*+    n: '15'+    expression: density+  number: '0.00002869621374274322137325433709257593165961052901982261502872546854856614255296226815568877765798637408'+  comment: $\Delta(A_{15}^{*})=\frac{84375\sqrt{15}\,\pi^{7}}{34394098106368}$+- params:+    family: A*+    n: '15'+    expression: centre+  number: '0.00007523061798547222103549939324306546194602063132948107993189367294286806783946073858887865431546088309'+  comment: $\delta(A_{15}^{*})=\frac{170859375\sqrt{15}}{8796093022208}$; $\det=\frac{1}{16}$,+    $\mu=\frac{15}{16}$, kissing number $32$+- params:+    family: A*+    n: '15'+    expression: hermite+  number: '1.127835971326984360272305102347244485327075963303601726962421015434996163709900834067412006867501465'+  comment: $\gamma(A_{15}^{*})=\frac{15}{2^{56/15}}$+- params:+    family: A*+    n: '16'+    expression: density+  number: '0.000009115730527441084092317713629740388914018878883339380598266389404523725588153261829420256293581249996'+  comment: $\Delta(A_{16}^{*})=\frac{512\sqrt{17}\,\pi^{8}}{2197363593915}$+- params:+    family: A*+    n: '16'+    expression: centre+  number: '0.00003873584375114728158351054057544847364196745562186837350337692882035961376851181840819029593599784200'+  comment: $\delta(A_{16}^{*})=\frac{65536\sqrt{17}}{6975757441}$; $\det=\frac{1}{17}$,+    $\mu=\frac{16}{17}$, kissing number $34$+- params:+    family: A*+    n: '16'+    expression: hermite+  number: '1.123502695890729540774873028726349966698516820755169676831194169154014775336602344148221496644896615'+  comment: $\gamma(A_{16}^{*})=\frac{16}{17^{15/16}}$+- params:+    family: A*+    n: '17'+    expression: density+  number: '0.000002807286781408316638295640060373858120273725992651788442016232554446651885175297198895982283445718239'+  comment: $\Delta(A_{17}^{*})=\frac{410338673\sqrt{17}\,\pi^{8}}{5718460310160998400}$+- params:+    family: A*+    n: '17'+    expression: centre+  number: '0.00001991250347508113859352333466441257379435276340674603892331878750254765006236882888081508594596403967'+  comment: $\delta(A_{17}^{*})=\frac{6975757441\sqrt{17}}{1444408272617472}$; $\det=\frac{1}{18}$,+    $\mu=\frac{17}{18}$, kissing number $36$+- params:+    family: A*+    n: '17'+    expression: hermite+  number: '1.119479062427896791535303203006780044804589499279177499125553892945337408812166626662943414206350314'+  comment: $\gamma(A_{17}^{*})=\frac{17}{2^{16/17}\cdot 3^{32/17}}$+- params:+    family: A*+    n: '18'+    expression: density+  number: '8.396394534243116942750547463458954113069587476047609566612383003759869251177299670009908576426216291e-7'+  comment: $\Delta(A_{18}^{*})=\frac{4782969\sqrt{19}\,\pi^{9}}{740168133657559040}$+- params:+    family: A*+    n: '18'+    expression: centre+  number: '0.00001022132072661282224937805245426146495562922065356840654269212382908442982738599300089075203223598147'+  comment: $\delta(A_{18}^{*})=\frac{387420489\sqrt{19}}{165216101262848}$; $\det=\frac{1}{19}$,+    $\mu=\frac{18}{19}$, kissing number $38$+- params:+    family: A*+    n: '18'+    expression: hermite+  number: '1.115734265234866944854986098392080008053192398765907077007752270042189707449356098498242413059114224'+  comment: $\gamma(A_{18}^{*})=\frac{18}{19^{17/18}}$+- params:+    family: A*+    n: '19'+    expression: density+  number: '2.442904637657317290592268067367763165544908550596852830245785646369921430225902433581091123304839847e-7'+  comment: $\Delta(A_{19}^{*})=\frac{16983563041\sqrt{19}\,\pi^{9}}{9033331507200000000000}$+- params:+    family: A*+    n: '19'+    expression: centre+  number: '0.000005239855740004983735897934480736998892496144036132633328886814777991680265976657679146434666937127839'+  comment: $\delta(A_{19}^{*})=\frac{322687697779\sqrt{19}}{268435456000000000}$;+    $\det=\frac{1}{20}$, $\mu=\frac{19}{20}$, kissing number $40$+- params:+    family: A*+    n: '19'+    expression: hermite+  number: '1.112240918036640366594758897653426845182016995816149071438909133396902229961636782708622281822396044'+  comment: $\gamma(A_{19}^{*})=\frac{19}{2^{36/19}\cdot 5^{18/19}}$+- params:+    family: A*+    n: '20'+    expression: density+  number: '6.923926315359767924173009961248276615178634727633231369760281101421982357598236884605994803319581915e-8'+  comment: $\Delta(A_{20}^{*})=\frac{390625\sqrt{21}\,\pi^{10}}{2421118083747831552}$+- params:+    family: A*+    n: '20'+    expression: centre+  number: '0.000002682975725638529010340702806732677732185527920817423622944184459279912668206851494848007254405465140'+  comment: $\delta(A_{20}^{*})=\frac{9765625\sqrt{21}}{16679880978201}$; $\det=\frac{1}{21}$,+    $\mu=\frac{20}{21}$, kissing number $42$+- params:+    family: A*+    n: '20'+    expression: hermite+  number: '1.108974769814499329717170067655231745318239781511967589118407581131761818718317214575653622775611009'+  comment: $\gamma(A_{20}^{*})=\frac{20}{3^{19/20}\cdot 7^{19/20}}$+- params:+    family: A*+    n: '21'+    expression: density+  number: '1.914227868615882773291375972943886863704713257552811289520862749058894629437533674465434555266016576e-8'+  comment: $\Delta(A_{21}^{*})=\frac{1400846643\sqrt{21}\,\pi^{10}}{31405472609614928281600}$+- params:+    family: A*+    n: '21'+    expression: centre+  number: '0.000001372289933427013491866991487965588803340410888367382136732905824436814799589710791801905117813354906'+  comment: $\delta(A_{21}^{*})=\frac{16679880978201\sqrt{21}}{55700195201880424448}$;+    $\det=\frac{1}{22}$, $\mu=\frac{21}{22}$, kissing number $44$+- params:+    family: A*+    n: '21'+    expression: hermite+  number: '1.105914375737187161985183787204636436466604835115040378925990157963012976242477446872005267850159509'+  comment: $\gamma(A_{21}^{*})=\frac{21}{2^{20/21}\cdot 11^{20/21}}$+- params:+    family: A*+    n: '22'+    expression: density+  number: '5.168210426133794663051746162163299634697400905956595971034585545256761555050117010062911555681944705e-9'+  comment: $\Delta(A_{22}^{*})=\frac{25937424601\sqrt{23}\,\pi^{11}}{7081074789412983393484800}$+- params:+    family: A*+    n: '22'+    expression: centre+  number: '7.012087168578560871576672144298089894766063803032390222252121468217926366300087090203903323415541361e-7'+  comment: $\delta(A_{22}^{*})=\frac{285311670611\sqrt{23}}{1951354384207722496}$;+    $\det=\frac{1}{23}$, $\mu=\frac{22}{23}$, kissing number $46$+- params:+    family: A*+    n: '22'+    expression: hermite+  number: '1.103040769308036072443570895291827614075395723503595137190776423705099269320424594870338528499041089'+  comment: $\gamma(A_{22}^{*})=\frac{22}{23^{21/22}}$+- params:+    family: A*+    n: '23'+    expression: density+  number: '1.364131836064964407869667097883241434002487344746052085782122991542884838368745210742899466835728432e-9'+  comment: $\Delta(A_{23}^{*})=\frac{41426511213649\sqrt{23}\,\pi^{11}}{42848392465514395024647782400}$+- params:+    family: A*+    n: '23'+    expression: centre+  number: '3.579781794998931904465208952879284491084851802739349122313187591048840675574681701060438665171695652e-7'+  comment: $\delta(A_{23}^{*})=\frac{952809757913927\sqrt{23}}{12764786611036820078592}$;+    $\det=\frac{1}{24}$, $\mu=\frac{23}{24}$, kissing number $48$+- params:+    family: A*+    n: '23'+    expression: hermite+  number: '1.100337156857078421258574319957845798641363442793016335968019213688914877163919775293714133938567575'+  comment: $\gamma(A_{23}^{*})=\frac{23}{2^{66/23}\cdot 3^{22/23}}$+- params:+    family: A*+    n: '24'+    expression: density+  number: '3.523436209156009968348354384146930874597117849821738316520513757221591158598065996616521976666113545e-10'+  comment: $\Delta(A_{24}^{*})=\frac{8748\pi^{12}}{22947788238525390625}$+- params:+    family: A*+    n: '24'+    expression: centre+  number: 2176782336/11920928955078125+  comment: $\delta(A_{24}^{*})=\frac{2176782336}{11920928955078125}$; $\det=\frac{1}{25}$,+    $\mu=\frac{24}{25}$, kissing number $50$+- params:+    family: A*+    n: '24'+    expression: hermite+  number: '1.097788642639603496857716303604875741959840539496487698611056199363117020686748520427747245299641720'+  comment: $\gamma(A_{24}^{*})=\frac{24}{5^{23/12}}$+- params:+    family: D*+    n: '5'+    expression: density+  number: '0.3289868133696452872944830333292050378437899802413596875471116458740014940806401747667257801239517411'+  comment: $\Delta(D_{5}^{*})=\frac{\pi^{2}}{30}$+- params:+    family: D*+    n: '5'+    expression: centre+  number: 1/16+  comment: $\delta(D_{5}^{*})=\frac{1}{16}$; $\det=\frac{1}{4}$, $\mu=1$, kissing+    number $10$+- params:+    family: D*+    n: '5'+    expression: hermite+  number: '1.319507910772894259374001971229640133033469013193418681505807795980535980893520830050353795235959499'+  comment: $\gamma(D_{5}^{*})=2^{2/5}$+- params:+    family: D*+    n: '6'+    expression: density+  number: '0.1614910243765615634139391409744864333449233779473182692403361359573249735284672189456599012114733656'+  comment: $\Delta(D_{6}^{*})=\frac{\pi^{3}}{192}$+- params:+    family: D*+    n: '6'+    expression: centre+  number: 1/32+  comment: $\delta(D_{6}^{*})=\frac{1}{32}$; $\det=\frac{1}{4}$, $\mu=1$, kissing+    number $12$+- params:+    family: D*+    n: '6'+    expression: hermite+  number: '1.259921049894873164767210607278228350570251464701507980081975112155299676513959483729396562436255094'+  comment: $\gamma(D_{6}^{*})=2^{1/3}$+- params:+    family: D*+    n: '7'+    expression: density+  number: '0.07382446828642814327494360730262236952910782991877406593843937643763427361301358580373024055381639570'+  comment: $\Delta(D_{7}^{*})=\frac{\pi^{3}}{420}$+- params:+    family: D*+    n: '7'+    expression: centre+  number: 1/64+  comment: $\delta(D_{7}^{*})=\frac{1}{64}$; $\det=\frac{1}{4}$, $\mu=1$, kissing+    number $14$+- params:+    family: D*+    n: '7'+    expression: hermite+  number: '1.219013654204475440911691002592560857277411935859960806590971514832067295459667993817258140313705125'+  comment: $\gamma(D_{7}^{*})=2^{2/7}$+- params:+    family: D*+    n: '8'+    expression: density+  number: '0.03170868848763100170457042079710452840160403179449395237352469381183947117735260292647196199047948976'+  comment: $\Delta(D_{8}^{*})=\frac{\pi^{4}}{3072}$+- params:+    family: D*+    n: '8'+    expression: centre+  number: 1/128+  comment: $\delta(D_{8}^{*})=\frac{1}{128}$; $\det=\frac{1}{4}$, $\mu=1$, kissing+    number $16$+- params:+    family: D*+    n: '8'+    expression: hermite+  number: '1.189207115002721066717499970560475915292972092463817413019002224719466668226917159870781344538137674'+  comment: $\gamma(D_{8}^{*})=2^{1/4}$+- params:+    family: D*+    n: '9'+    expression: density+  number: '0.01288480040132307370852385353025199884255655895141341556765447875528715019270201007805844804692499901'+  comment: $\Delta(D_{9}^{*})=\frac{\pi^{4}}{7560}$+- params:+    family: D*+    n: '9'+    expression: centre+  number: 1/256+  comment: $\delta(D_{9}^{*})=\frac{1}{256}$; $\det=\frac{1}{4}$, $\mu=1$, kissing+    number $18$+- params:+    family: D*+    n: '9'+    expression: hermite+  number: '1.166529039576116580893692634660668061417909153898898354909265761772354220490178139945062540827940208'+  comment: $\gamma(D_{9}^{*})=2^{2/9}$+- params:+    family: D*+    n: '10'+    expression: density+  number: '0.004980789140385440320031596843154876407556977646941324716039916075622314325091374096448299161352622378'+  comment: $\Delta(D_{10}^{*})=\frac{\pi^{5}}{61440}$+- params:+    family: D*+    n: '10'+    expression: centre+  number: 1/512+  comment: $\delta(D_{10}^{*})=\frac{1}{512}$; $\det=\frac{1}{4}$, $\mu=1$, kissing+    number $20$+- params:+    family: D*+    n: '10'+    expression: hermite+  number: '1.148698354997035006798626946777927589443850889097797505513711118493603206253513056811473113011508474'+  comment: $\gamma(D_{10}^{*})=2^{1/5}$+- params:+    family: D*+    n: '11'+    expression: density+  number: '0.001839945194716699454441686568322724906687714686315987196690070007733495623698978021198794495391444919'+  comment: $\Delta(D_{11}^{*})=\frac{\pi^{5}}{166320}$+- params:+    family: D*+    n: '11'+    expression: centre+  number: 1/1024+  comment: $\delta(D_{11}^{*})=\frac{1}{1024}$; $\det=\frac{1}{4}$, $\mu=1$, kissing+    number $22$+- params:+    family: D*+    n: '11'+    expression: hermite+  number: '1.134312522195462580992497732917119138938301920796695962016192664318287105778049009048037048416301812'+  comment: $\gamma(D_{11}^{*})=2^{2/11}$+- params:+    family: D*+    n: '12'+    expression: density+  number: '0.0006519837738547800272828636635012613246441091092129493651944147321805107373334080766460648662325817438'+  comment: $\Delta(D_{12}^{*})=\frac{\pi^{6}}{1474560}$+- params:+    family: D*+    n: '12'+    expression: centre+  number: 1/2048+  comment: $\delta(D_{12}^{*})=\frac{1}{2048}$; $\det=\frac{1}{4}$, $\mu=1$, kissing+    number $24$+- params:+    family: D*+    n: '12'+    expression: hermite+  number: '1.122462048309372981433533049679179516232411110613986753440409545882904005565861247087923227112509081'+  comment: $\gamma(D_{12}^{*})=2^{1/6}$+- params:+    family: D*+    n: '13'+    expression: density+  number: '0.0002223214733357624868257250720696941713072153605841026140389879073436040609488544357261306769970575110'+  comment: $\Delta(D_{13}^{*})=\frac{\pi^{6}}{4324320}$+- params:+    family: D*+    n: '13'+    expression: centre+  number: 1/4096+  comment: $\delta(D_{13}^{*})=\frac{1}{4096}$; $\det=\frac{1}{4}$, $\mu=1$, kissing+    number $26$+- params:+    family: D*+    n: '13'+    expression: hermite+  number: '1.112531476096486920262088977149391353845053178180482097211804844090556252100582416200587768880471251'+  comment: $\gamma(D_{13}^{*})=2^{2/13}$+- params:+    family: D*+    n: '14'+    expression: density+  number: '0.00007315240836435450157321037396802246655460981278907902460865407047291430923930847848410113193790519922'+  comment: $\Delta(D_{14}^{*})=\frac{\pi^{7}}{41287680}$+- params:+    family: D*+    n: '14'+    expression: centre+  number: 1/8192+  comment: $\delta(D_{14}^{*})=\frac{1}{8192}$; $\det=\frac{1}{4}$, $\mu=1$, kissing+    number $28$+- params:+    family: D*+    n: '14'+    expression: hermite+  number: '1.104089513673812337649505387623344721325326600780124165514532464142106322880380980716598289886302005'+  comment: $\gamma(D_{14}^{*})=2^{1/7}$+- params:+    family: D*+    n: '15'+    expression: density+  number: '0.00002328145024556301775010642515718881297650985184025389936262992017537350510056002547559271456236672075'+  comment: $\Delta(D_{15}^{*})=\frac{\pi^{7}}{129729600}$+- params:+    family: D*+    n: '15'+    expression: centre+  number: 1/16384+  comment: $\delta(D_{15}^{*})=\frac{1}{16384}$; $\det=\frac{1}{4}$, $\mu=1$, kissing+    number $30$+- params:+    family: D*+    n: '15'+    expression: hermite+  number: '1.096824979694625960681471778028803387827847321527142350042026590511525718728112099321909227103947672'+  comment: $\gamma(D_{15}^{*})=2^{2/15}$+- params:+    family: D*+    n: '16'+    expression: density+  number: '0.000007181720897183020158138407982649701585237699441789508011310888786215183395681968791927961566727437766'+  comment: $\Delta(D_{16}^{*})=\frac{\pi^{8}}{1321205760}$+- params:+    family: D*+    n: '16'+    expression: centre+  number: 1/32768+  comment: $\delta(D_{16}^{*})=\frac{1}{32768}$; $\det=\frac{1}{4}$, $\mu=1$, kissing+    number $32$+- params:+    family: D*+    n: '16'+    expression: hermite+  number: '1.090507732665257659207010655760707978992702718540067121785667647683300530848841840338211140494203120'+  comment: $\gamma(D_{16}^{*})=2^{1/8}$+- params:+    family: D*+    n: '17'+    expression: density+  number: '0.000002151200972246384245549425044796064002423044337570808524289366093027095658025565641701133000763514610'+  comment: $\Delta(D_{17}^{*})=\frac{\pi^{8}}{4410806400}$+- params:+    family: D*+    n: '17'+    expression: centre+  number: 1/65536+  comment: $\delta(D_{17}^{*})=\frac{1}{65536}$; $\det=\frac{1}{4}$, $\mu=1$, kissing+    number $34$+- params:+    family: D*+    n: '17'+    expression: hermite+  number: '1.084963913643637129667077523875454847611686369477650775238069264432332883787604319563593590399253638'+  comment: $\gamma(D_{17}^{*})=2^{2/17}$+- params:+    family: D*+    n: '18'+    expression: density+  number: '6.267233780756243041900815240760676941515919555492639397959052365893117299557815407429373434731789822e-7'+  comment: $\Delta(D_{18}^{*})=\frac{\pi^{9}}{47563407360}$+- params:+    family: D*+    n: '18'+    expression: centre+  number: 1/131072+  comment: $\delta(D_{18}^{*})=\frac{1}{131072}$; $\det=\frac{1}{4}$, $\mu=1$, kissing+    number $36$+- params:+    family: D*+    n: '18'+    expression: hermite+  number: '1.080059738892306169872930831288596912737466762465644733136497122283534876578258927885325469533216098'+  comment: $\gamma(D_{18}^{*})=2^{1/9}$+- params:+    family: D*+    n: '19'+    expression: density+  number: '1.778472939685384589378474648156451640170548416240947479203466002083488862472974043782248662749063967e-7'+  comment: $\Delta(D_{19}^{*})=\frac{\pi^{9}}{167610643200}$+- params:+    family: D*+    n: '19'+    expression: centre+  number: 1/262144+  comment: $\delta(D_{19}^{*})=\frac{1}{262144}$; $\det=\frac{1}{4}$, $\mu=1$, kissing+    number $38$+- params:+    family: D*+    n: '19'+    expression: hermite+  number: '1.075690586220182474231400776464092135133759903355640199420412371622904977065515069260940075713230671'+  comment: $\gamma(D_{19}^{*})=2^{2/19}$+- params:+    family: D*+    n: '20'+    expression: density+  number: '4.922273900988399508018168306903619700848854264859521101341700080483796946706343814000842714927006742e-8'+  comment: $\Delta(D_{20}^{*})=\frac{\pi^{10}}{1902536294400}$+- params:+    family: D*+    n: '20'+    expression: centre+  number: 1/524288+  comment: $\delta(D_{20}^{*})=\frac{1}{524288}$; $\det=\frac{1}{4}$, $\mu=1$, kissing+    number $40$+- params:+    family: D*+    n: '20'+    expression: hermite+  number: '1.071773462536293164213006325023342022906384604977556783482780668144245438837468954516907443983687770'+  comment: $\gamma(D_{20}^{*})=2^{1/10}$+- params:+    family: D*+    n: '21'+    expression: density+  number: '1.330294648077106588591918620744521198986072804808539550841437845846974972187139787743151495214615370e-8'+  comment: $\Delta(D_{21}^{*})=\frac{\pi^{10}}{7039647014400}$+- params:+    family: D*+    n: '21'+    expression: centre+  number: 1/1048576+  comment: $\delta(D_{21}^{*})=\frac{1}{1048576}$; $\det=\frac{1}{4}$, $\mu=1$, kissing+    number $42$+- params:+    family: D*+    n: '21'+    expression: hermite+  number: '1.068241690814402220013606897069232472303291177670581722176337650608827204480478972033118520513530295'+  comment: $\gamma(D_{21}^{*})=2^{2/21}$+- params:+    family: D*+    n: '22'+    expression: density+  number: '3.514495346886802090291542986531630278211421125421999669027253491314644897376107195514171767354529458e-9'+  comment: $\Delta(D_{22}^{*})=\frac{\pi^{11}}{83711596953600}$+- params:+    family: D*+    n: '22'+    expression: centre+  number: 1/2097152+  comment: $\delta(D_{22}^{*})=\frac{1}{2097152}$; $\det=\frac{1}{4}$, $\mu=1$, kissing+    number $44$+- params:+    family: D*+    n: '22'+    expression: hermite+  number: '1.065041089439962678190592595398204490092328886510584811019914747456601900170422889132828048742624901'+  comment: $\gamma(D_{22}^{*})=2^{1/11}$+- params:+    family: D*+    n: '23'+    expression: density+  number: '9.085312811975777010472139868275328175361231916133286997125711576079548388754319841122253240630454458e-10'+  comment: $\Delta(D_{23}^{*})=\frac{\pi^{11}}{323823762662400}$+- params:+    family: D*+    n: '23'+    expression: centre+  number: 1/4194304+  comment: $\delta(D_{23}^{*})=\frac{1}{4194304}$; $\det=\frac{1}{4}$, $\mu=1$, kissing+    number $46$+- params:+    family: D*+    n: '23'+    expression: hermite+  number: '1.062127176862690915800524153411598751007243459063598140151676822461795332663241154793066281672813239'+  comment: $\gamma(D_{23}^{*})=2^{2/23}$+- params:+    family: D*+    n: '24'+    expression: density+  number: '2.300231825594810304526502565963217783175077217817668142598447343506211798490268625601944563610725300e-10'+  comment: $\Delta(D_{24}^{*})=\frac{\pi^{12}}{4018156653772800}$+- params:+    family: D*+    n: '24'+    expression: centre+  number: 1/8388608+  comment: $\delta(D_{24}^{*})=\frac{1}{8388608}$; $\det=\frac{1}{4}$, $\mu=1$, kissing+    number $48$+- params:+    family: D*+    n: '24'+    expression: hermite+  number: '1.059463094359295264561825294946341700779204317494185628559208431458761646063255722383768376863945569'+  comment: $\gamma(D_{24}^{*})=2^{1/12}$+- params:+    family: E*+    n: '6'+    expression: density+  number: '0.3315089294062799462786467188266294279054798197941221005434702805309183751374961951311348636758582602'+  comment: $\Delta(E_{6}^{*})=\frac{\sqrt{3}\,\pi^{3}}{162}$+- params:+    family: E*+    n: '6'+    expression: centre+  number: '0.06415002990995841827879430894466193951640019458556965289095581405377529692254814952152393284286105836'+  comment: $\delta(E_{6}^{*})=\frac{\sqrt{3}}{27}$; $\det=\frac{1}{3}$, $\mu=\frac{4}{3}$,+    kissing number $54$+- params:+    family: E*+    n: '6'+    expression: hermite+  number: '1.601249273568003635567287183132723795064543224301309633357828520598940805358792781392989051159671462'+  comment: $\gamma(E_{6}^{*})=\frac{4}{3^{5/6}}$+- params:+    family: E*+    n: '7'+    expression: density+  number: '0.2157767942296232864617263018076900472706203469909955458001694950955709781028881484380333175175898855'+  comment: $\Delta(E_{7}^{*})=\frac{9\sqrt{3}\,\pi^{3}}{2240}$+- params:+    family: E*+    n: '7'+    expression: centre+  number: '0.04566930840269500676293071408267436905024974790320339546631522309101776509427500097772553422113839018'+  comment: $\delta(E_{7}^{*})=\frac{27\sqrt{3}}{1024}$; $\det=\frac{1}{2}$, $\mu=\frac{3}{2}$,+    kissing number $56$+- params:+    family: E*+    n: '7'+    expression: hermite+  number: '1.656134270510718506474258081435017081987989901170186248271798696213159484320571471074897434829453008'+  comment: $\gamma(E_{7}^{*})=\frac{3}{2^{6/7}}$+- params:+    family: Lambda+    n: '9'+    expression: density+  number: '0.1457748758081711105332137544576519995425739925107256670963211484639631773090752900724100031467048256'+  comment: $\Delta(\Lambda_{9})=\frac{\sqrt{2}\,\pi^{4}}{945}$; the densest lattice+    packing known in dimension 9 CITE{Catalogue-density}+- params:+    family: Lambda+    n: '9'+    expression: centre+  number: '0.04419417382415922027505277263155306495530224610552962728677124181221038995194084496407461044773879915'+  comment: $\delta(\Lambda_{9})=\frac{\sqrt{2}}{32}$; $\det=512$, $\mu=4$, kissing+    number $272$+- params:+    family: Lambda+    n: '9'+    expression: hermite+  number: '2'+  comment: $\gamma(\Lambda_{9})=2$+- params:+    family: Lambda+    n: '10'+    expression: density+  number: '0.09202111843130555779386059577030574946623950376510629227349760612731057277995423712397819980248230119'+  comment: $\Delta(\Lambda_{10})=\frac{\sqrt{3}\,\pi^{5}}{5760}$; the densest lattice+    packing known in dimension 10 CITE{Catalogue-density}+- params:+    family: Lambda+    n: '10'+    expression: centre+  number: '0.03608439182435161028182179878137234097797510945438292975116264540524860451893333410585721222410934533'+  comment: $\delta(\Lambda_{10})=\frac{\sqrt{3}}{48}$; $\det=768$, $\mu=4$, kissing+    number $336$+- params:+    family: Lambda+    n: '10'+    expression: hermite+  number: '2.058372017929521168115134100140546258980859052190904222384855679777514164955808108505955673298293323'+  comment: $\gamma(\Lambda_{10})=\frac{2^{6/5}}{3^{1/10}}$+- params:+    family: Lambda+    n: '11'+    expression: density+  number: '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 $\frac{\sqrt{3}}{54}$'+- params:+    family: Lambda+    n: '11'+    expression: centre+  number: 1/32+  comment: $\delta(\Lambda_{11})=\frac{1}{32}$; $\det=1024$, $\mu=4$, kissing number+    $438$+- params:+    family: Lambda+    n: '11'+    expression: hermite+  number: '2.130082178879925356381185190796408980184657773021169622039829494913203800340845778265656097485249802'+  comment: $\gamma(\Lambda_{11})=2^{12/11}$+- params:+    family: Lambda+    n: '12'+    expression: density+  number: '0.04172696152670592174610327446408072477722298298962875937244254285955268718933811690534815143888523160'+  comment: '$\Delta(\Lambda_{12})=\frac{\pi^{6}}{23040}$; not the densest lattice+    packing known in dimension 12: $K_{12}$ has centre density $\frac{1}{27}$'+- params:+    family: Lambda+    n: '12'+    expression: centre+  number: 1/32+  comment: $\delta(\Lambda_{12})=\frac{1}{32}$; $\det=1024$, $\mu=4$, kissing number+    $648$+- params:+    family: Lambda+    n: '12'+    expression: hermite+  number: '2.244924096618745962867066099358359032464822221227973506880819091765808011131722494175846454225018161'+  comment: $\gamma(\Lambda_{12})=2^{7/6}$+- params:+    family: Lambda+    n: '13'+    expression: density+  number: '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 $\frac{\sqrt{3}}{54}$'+- params:+    family: Lambda+    n: '13'+    expression: centre+  number: 1/32+  comment: $\delta(\Lambda_{13})=\frac{1}{32}$; $\det=1024$, $\mu=4$, kissing number+    $906$+- params:+    family: Lambda+    n: '13'+    expression: hermite+  number: '2.346920920009252782190883505970653978257647218823348190514280236112778699531056704530044027110891233'+  comment: $\gamma(\Lambda_{13})=2^{16/13}$+- params:+    family: Lambda+    n: '14'+    expression: density+  number: '0.02162409608244710546249221367868409564177065936712948464424487710130327456120990167726026005937182741'+  comment: $\Delta(\Lambda_{14})=\frac{\sqrt{3}\,\pi^{7}}{241920}$; the densest lattice+    packing known in dimension 14 CITE{Catalogue-density}+- params:+    family: Lambda+    n: '14'+    expression: centre+  number: '0.03608439182435161028182179878137234097797510945438292975116264540524860451893333410585721222410934533'+  comment: $\delta(\Lambda_{14})=\frac{\sqrt{3}}{48}$; $\det=768$, $\mu=4$, kissing+    number $1422$+- params:+    family: Lambda+    n: '14'+    expression: hermite+  number: '2.488643919822375497998830713027331282995624292467588082920749685538170938078947312586119317650029594'+  comment: $\gamma(\Lambda_{14})=\frac{2^{10/7}}{3^{1/14}}$+- params:+    family: Lambda+    n: '15'+    expression: density+  number: '0.01685757065676269764781550358540912002543295423644778555552670648966997295106061582291034513965069853'+  comment: $\Delta(\Lambda_{15})=\frac{8\sqrt{2}\,\pi^{7}}{2027025}$; the densest+    lattice packing known in dimension 15 CITE{Catalogue-density}+- params:+    family: Lambda+    n: '15'+    expression: centre+  number: '0.04419417382415922027505277263155306495530224610552962728677124181221038995194084496407461044773879915'+  comment: $\delta(\Lambda_{15})=\frac{\sqrt{2}}{32}$; $\det=512$, $\mu=4$, kissing+    number $2340$+- params:+    family: Lambda+    n: '15'+    expression: hermite+  number: '2.639015821545788518748003942459280266066938026386837363011615591961071961787041660100707590471918997'+  comment: $\gamma(\Lambda_{15})=2^{7/5}$+- params:+    family: Lambda+    n: '16'+    expression: density+  number: '0.01470816439743082528386745954846658884656680845678491240716470023416869559435667208586846528865779255'+  comment: $\Delta(\Lambda_{16})=\frac{\pi^{8}}{645120}$; the densest lattice packing+    known in dimension 16 CITE{Catalogue-density}; $\Lambda_{16}=BW_{16}$, the Barnes–Wall+    lattice+- params:+    family: Lambda+    n: '16'+    expression: centre+  number: 1/16+  comment: $\delta(\Lambda_{16})=\frac{1}{16}$; $\det=256$, $\mu=4$, kissing number+    $4320$; $\Lambda_{16}=BW_{16}$, the Barnes–Wall lattice+- params:+    family: Lambda+    n: '16'+    expression: hermite+  number: '2.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145'+  comment: $\gamma(\Lambda_{16})=2^{3/2}$; $\Lambda_{16}=BW_{16}$, the Barnes–Wall+    lattice+- params:+    family: Lambda+    n: '17'+    expression: density+  number: '0.008811319182321189869770444983484678153924789606690031715489243517038983815272716868407840771127355841'+  comment: $\Delta(\Lambda_{17})=\frac{32\pi^{8}}{34459425}$; the densest lattice+    packing known in dimension 17 CITE{Catalogue-density}+- params:+    family: Lambda+    n: '17'+    expression: centre+  number: 1/16+  comment: $\delta(\Lambda_{17})=\frac{1}{16}$; $\det=256$, $\mu=4$, kissing number+    $5346$+- params:+    family: Lambda+    n: '17'+    expression: hermite+  number: '2.886681154059912883177543043257111681405167207157733637364231605191669617520219793568933482805778991'+  comment: $\gamma(\Lambda_{17})=2^{26/17}$+- params:+    family: Lambda+    n: '18'+    expression: density+  number: '0.005928368718469419730049891486523909806100960042824357061950447334418422452973803955425827012554145075'+  comment: $\Delta(\Lambda_{18})=\frac{\sqrt{3}\,\pi^{9}}{8709120}$; the densest lattice+    packing known in dimension 18 CITE{Catalogue-density}+- params:+    family: Lambda+    n: '18'+    expression: centre+  number: '0.07216878364870322056364359756274468195595021890876585950232529081049720903786666821171442444821869065'+  comment: $\delta(\Lambda_{18})=\frac{\sqrt{3}}{24}$; $\det=192$, $\mu=4$, kissing+    number $7398$+- params:+    family: Lambda+    n: '18'+    expression: hermite+  number: '2.986825999361043925804195343609699263561550450174920817714066244307066740702547366886684630008909785'+  comment: $\gamma(\Lambda_{18})=\frac{2^{5/3}}{3^{1/18}}$+- params:+    family: Lambda+    n: '19'+    expression: density+  number: '0.004120806279768667500886821058357660528632336189215187578663362443680168202332562149057259835876284349'+  comment: $\Delta(\Lambda_{19})=\frac{64\sqrt{2}\,\pi^{9}}{654729075}$; the densest+    lattice packing known in dimension 19 CITE{Catalogue-density}+- params:+    family: Lambda+    n: '19'+    expression: centre+  number: '0.08838834764831844055010554526310612991060449221105925457354248362442077990388168992814922089547759830'+  comment: $\delta(\Lambda_{19})=\frac{\sqrt{2}}{16}$; $\det=128$, $\mu=4$, kissing+    number $10668$+- params:+    family: Lambda+    n: '19'+    expression: hermite+  number: '3.098519284533311451212543035603137070013123293883257942512309292804881953057023682104046634319766642'+  comment: $\gamma(\Lambda_{19})=2^{31/19}$+- params:+    family: Lambda+    n: '20'+    expression: density+  number: '0.003225861423751757501574786781612356207148305131018335748975296564745861166993469481943592281654563139'+  comment: $\Delta(\Lambda_{20})=\frac{\pi^{10}}{29030400}$; the densest lattice packing+    known in dimension 20 CITE{Catalogue-density}+- params:+    family: Lambda+    n: '20'+    expression: centre+  number: 1/8+  comment: $\delta(\Lambda_{20})=\frac{1}{8}$; $\det=64$, $\mu=4$, kissing number+    $17400$+- params:+    family: Lambda+    n: '20'+    expression: hermite+  number: '3.249009585424942090438837531101126605140819897728637770129974852216882973740059542498944519264115310'+  comment: $\gamma(\Lambda_{20})=2^{17/10}$+- params:+    family: Lambda+    n: '21'+    expression: density+  number: '0.002465884711502463245944355824165164696567691388297833979591321081883716334406532970306523662963463407'+  comment: $\Delta(\Lambda_{21})=\frac{256\sqrt{2}\,\pi^{10}}{13749310575}$; the densest+    lattice packing known in dimension 21 CITE{Catalogue-density}+- params:+    family: Lambda+    n: '21'+    expression: centre+  number: '0.1767766952966368811002110905262122598212089844221185091470849672488415598077633798562984417909551966'+  comment: $\delta(\Lambda_{21})=\frac{\sqrt{2}}{8}$; $\det=32$, $\mu=4$, kissing+    number $27720$+- params:+    family: Lambda+    n: '21'+    expression: hermite+  number: '3.391455967510140347881421973682253209168920269455297005724336499334965922559574004478846549769583126'+  comment: $\gamma(\Lambda_{21})=2^{37/21}$+- params:+    family: Lambda+    n: '22'+    expression: density+  number: '0.002127660145275864210626933444611565330655697515338490683084571822978614708729444826368239332278713761'+  comment: $\Delta(\Lambda_{22})=\frac{\sqrt{3}\,\pi^{11}}{239500800}$; the densest+    lattice packing known in dimension 22 CITE{Catalogue-density}+- params:+    family: Lambda+    n: '22'+    expression: centre+  number: '0.2886751345948128822545743902509787278238008756350634380093011632419888361514666728468576977928747626'+  comment: $\delta(\Lambda_{22})=\frac{\sqrt{3}}{6}$; $\det=12$, $\mu=4$, kissing+    number $49896$+- params:+    family: Lambda+    n: '22'+    expression: hermite+  number: '3.572780195142164825133666976858288632976880296572663439444579656875091486429634301963404042025028554'+  comment: $\gamma(\Lambda_{22})=\frac{2^{21/11}}{3^{1/22}}$+- params:+    family: Lambda+    n: '23'+    expression: density+  number: '0.001905328193426062470906566906903334103361515823538275509259618028319837706257289936344921562809463883'+  comment: $\Delta(\Lambda_{23})=\frac{2048\pi^{11}}{316234143225}$; the densest lattice+    packing known in dimension 23 CITE{Catalogue-density}+- params:+    family: Lambda+    n: '23'+    expression: centre+  number: 1/2+  comment: $\delta(\Lambda_{23})=\frac{1}{2}$; $\det=4$, $\mu=4$, kissing number $93150$+- params:+    family: Lambda+    n: '23'+    expression: hermite+  number: '3.766027352595563767271038746597904900756830978894098804583035617966410295461698780635574455788248708'+  comment: $\gamma(\Lambda_{23})=2^{44/23}$+- params:+    family: Lambda+    n: '24'+    expression: density+  number: '0.001929574309403923047903345563685957640168471815000303352234647617331495634250985531487347698186143913'+  comment: $\Delta(\Lambda_{24})=\frac{\pi^{12}}{479001600}$; the densest packing+    of any kind in dimension 24 CITE{CKMRV}; $\Lambda_{24}$ is the Leech lattice+- params:+    family: Lambda+    n: '24'+    expression: centre+  number: '1'+  comment: $\delta(\Lambda_{24})=1$; $\det=1$, $\mu=4$, kissing number $196560$; $\Lambda_{24}$+    is the Leech lattice+  equals: HREF{One}+- params:+    family: Lambda+    n: '24'+    expression: hermite+  number: '4'+  comment: $\gamma(\Lambda_{24})=4$; Hermite's constant $\gamma_{24}$ CITE{CohnKumar};+    $\Lambda_{24}$ is the Leech lattice+- params:+    family: K+    n: '12'+    expression: density+  number: '0.04945417662424405540278906603150308121744946132104149258956153227798837000217850892485706837201212635'+  comment: $\Delta(K_{12})=\frac{\pi^{6}}{19440}$; the densest lattice packing known+    in dimension 12 CITE{Catalogue-density}; $K_{12}$ is the Coxeter–Todd lattice+- params:+    family: K+    n: '12'+    expression: centre+  number: 1/27+  comment: $\delta(K_{12})=\frac{1}{27}$; $\det=729$, $\mu=4$, kissing number $756$;+    $K_{12}$ is the Coxeter–Todd lattice+- params:+    family: K+    n: '12'+    expression: hermite+  number: '2.309401076758503058036595122007829822590407005080507504074409305935910689211733382774861582342998101'+  comment: $\gamma(K_{12})=\frac{4}{\sqrt{3}}$; $K_{12}$ is the Coxeter–Todd lattice