On the Classification of Lattices Over <svg      style="vertical-align:-3.06pt;width:75px;" id="M1" height="24.4125" version="1.1" viewBox="0 0 75 24.4125" width="75"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.022,-0,0,-.022,26.138,16.937)"><path id="x221A" d="M782 703l-309 -950l-231 505l-91 -47l-18 35l147 78l194 -431h2l264 810h42z" /></g><g transform="matrix(.022,-0,0,-.022,.062,20.538)"><path id="x211A" d="M794 -138l-9 -21q-74 -1 -142 15q-79 19 -254 130q-139 7 -231 87q-52 44 -83 107t-31 135q0 145 103 247.5t274 102.5q145 0 249.5 -93.5t104.5 -240.5q0 -126 -73 -212.5t-179 -116.5q137 -85 271 -140zM673 302q0 152 -78 239.5t-201 87.5q-50 0 -84.5 -74t-34.5 -209
q0 -265 118 -317l66 18l31 -19q76 19 129.5 90t53.5 184zM298 54q-66 94 -66 292q0 178 50 256q-62 -32 -99 -98t-37 -162q0 -98 41 -174t111 -114z" /></g><g transform="matrix(.022,-0,0,-.022,18.377,20.538)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g>

			<g transform="matrix(1.25,0,0,-1.25,90,90)">
				<path d="m -37.39,70.61 19.13,0" style="fill:none;stroke:#000000;stroke-width:0.89700001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none" />
			</g>
			<g transform="matrix(.022,-0,0,-.022,43.262,20.538)"><path id="x2212" d="M535 230h-483v50h483v-50z" /></g><g transform="matrix(.022,-0,0,-.022,56.421,20.538)"><path id="x33" d="M285 378v-2q65 -13 102 -54.5t37 -97.5q0 -57 -30.5 -104.5t-74 -75t-85.5 -42t-72 -14.5q-31 0 -59.5 11t-40.5 23q-19 18 -16 36q1 16 23 33q13 10 24 0q58 -51 124 -51q55 0 88 40t33 112q0 64 -39 96.5t-88 32.5q-29 0 -64 -11l-6 29q77 25 118 57.5t41 84.5
q0 45 -26.5 69.5t-68.5 24.5q-67 0 -120 -79l-20 20l43 63q51 56 127 56h1q66 0 107 -37t41 -95q0 -42 -31 -71q-22 -23 -68 -54z" /></g><g transform="matrix(.022,-0,0,-.022,67.181,20.538)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g>

		
	
</svg> Which Are Even Unimodular <svg     style="vertical-align:-0.0pt;width:16.875px;" id="M2" height="15.25" version="1.1" viewBox="0 0 16.875 15.25" width="16.875"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.022,-0,0,-.022,.062,15.187)"><path id="x2124" d="M734 170q-25 -130 -36 -170h-661l-8 15l420 599h-173q-77 0 -102 -7t-39 -29q-19 -29 -35 -87l-29 1l12 183h24q9 -16 19 -20.5t34 -4.5h543l7 -14l-416 -598l166 -2q94 0 131.5 7.5t58.5 29.5q28 29 55 103zM639 614h-89l-397 -567l88 -7z" /></g>

		
	
</svg>-Lattices of Rank 32

<div class="floats-full-page floats-full-page-other"><h2>Table 1</h2><div class="floats-full-page-caption">The lattice of rank 4.</div><div class="floats-full-page-content"><table class="table-group" id="tab1"><tr><td><table class="table"><tr><td class="thead-hr" colspan="5"><hr /></td></tr><tr class="thead"><td align="left">no.</td><td align="center"><span style="width: 11.075px;"><svg height="11.3125" id="M92" style="vertical-align:-0.1092pt" version="1.1" viewbox="0 0 11.075 11.3125" width="11.075" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path d="M627 18l-10 -26q-79 6 -116 27t-69 76q-41 71 -71 138q-13 29 -27.5 39t-42.5 10h-46l-27 -145q-13 -74 -2.5 -88.5t78.5 -20.5l-6 -28h-271l5 28q66 6 82.5 21.5t30.5 87.5l71 387q12 66 2 78.5t-77 19.5l8 28h233q102 0 147 -29q65 -43 65 -129q0 -69 -45.5 -117&#10;t-115.5 -72q40 -86 66 -133q39 -68 65 -101q28 -37 73 -51zM491 483q0 67 -33.5 101t-91.5 34q-35 0 -51 -10q-13 -8 -20 -48l-45 -245h49q71 0 113 28q79 52 79 140z" id="x1D445"></path></g>
</svg></span></td><td align="center"><svg height="12.675" id="M93" style="vertical-align:-1.7199pt" version="1.1" viewbox="0 0 11.6375 12.675" width="11.6375" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,10.462)"><path d="M624 398l-8 -50h-120l-40 -185h112l-8 -50h-115l-52 -239h-59l51 239h-148l-52 -239h-59l53 239h-126l9 50h128l41 185h-121l8 50h124l47 214h59l-48 -214h149l46 214h59l-47 -214h117zM439 348h-148l-40 -185h147z" id="x23"></path></g>
</svg>Aut</td><td align="center"><span style="width: 39.712502px;"><svg height="17.4125" id="M94" style="vertical-align:-4.39398pt" version="1.1" viewbox="0 0 39.712502 17.4125" width="39.712502" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.875)"><path d="M1012 105l19 -16q-20 -28 -60 -62q-37 -33 -71 -49h-14q-98 35 -98 202q0 94 50 183q49 90 118 131v2q-78 13 -93 64q-2 6 -6 24t-7 25.5t-10 18.5t-18.5 15.5t-28.5 4.5q-33 0 -58 -30q-25 -28 -25 -69q9 -48 9 -116q0 -79 -10 -130q-11 -54 -45 -120&#10;q-60 -109 -147 -205q-13 8 -43 29q-35 25 -53 33q-23 -19 -45 -49l-22 14l94 145q1 -1 26.5 -19.5t45.5 -31.5t22 -11q52 0 52 323q0 159 -14 198t-61 39q-31 0 -55 -28t-24 -58v-123q0 -81 -10 -132q-9 -55 -45 -122q-59 -113 -147 -207l-36 21q-63 39 -76 41&#10;q-19 -10 -44 -49l-22 14l94 145q104 -62 110 -62q37 0 45 120q2 30 5 97q1 25 1 100v39q0 52 -8 95q-8 56 -28 87q-24 38 -63 38q-30 1 -48.5 -20t-18.5 -47t12 -47q9 -15 28.5 -31.5l35 -28.5t27.5 -35t12 -54q0 -50 -50 -94t-116 -60l-9 22q30 10 52.5 32t22.5 49&#10;q0 55 -57 94q-57 41 -57 96q0 72 53 113q54 42 132 42q55 0 104 -24.5t76 -66.5h2q18 37 55 62t77 25q99 0 142 -87h2q21 39 55.5 63t74.5 24q96 0 127 -67q5 -10 15 -40t11 -34q14 -35 51 -39v-14q-75 -35 -99 -106q-23 -72 -23 -183q0 -153 36 -153q27 2 71 50z" id="x1D510"></path></g>
<g transform="matrix(.012,-0,0,-.012,18.475,15.95)"><path d="M285 378v-2q65 -13 102 -54.5t37 -97.5q0 -57 -30.5 -104.5t-74 -75t-85.5 -42t-72 -14.5q-31 0 -59.5 11t-40.5 23q-19 18 -16 36q1 16 23 33q13 10 24 0q58 -51 124 -51q55 0 88 40t33 112q0 64 -39 96.5t-88 32.5q-29 0 -64 -11l-6 29q77 25 118 57.5t41 84.5&#10;q0 45 -26.5 69.5t-68.5 24.5q-67 0 -120 -79l-20 20l43 63q51 56 127 56h1q66 0 107 -37t41 -95q0 -42 -31 -71q-22 -23 -68 -54z" id="x33"></path></g><g transform="matrix(.012,-0,0,-.012,24.186,15.95)"><path d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z" id="x2C"></path></g><g transform="matrix(.012,-0,0,-.012,26.899,15.95)"><path d="M983 225q0 -112 -67 -174.5t-150 -62.5q-91 0 -154.5 43.5t-113.5 129.5q-49 -85 -104 -129t-138 -44q-98 0 -158.5 66t-60.5 154q0 59 21 106.5t54.5 75.5t70.5 43t73 15q90 0 152.5 -43.5t112.5 -128.5q48 84 104.5 128t140.5 44q93 0 155 -65t62 -158zM478 196&#10;q-27 49 -47 80t-50 67t-64 54t-73 18q-48 0 -81.5 -47t-33.5 -128q0 -96 37.5 -157.5t99.5 -61.5q68 0 117.5 47t94.5 128zM889 204q0 91 -35.5 151t-99.5 60q-68 0 -119 -47t-95 -127q27 -49 47 -80.5t50 -67.5t65 -54t74 -18q113 0 113 183z" id="x221E"></path></g>
</svg></span></td><td align="center"><span style="width: 39.712502px;"><svg height="17.4125" id="M95" style="vertical-align:-4.39398pt" version="1.1" viewbox="0 0 39.712502 17.4125" width="39.712502" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.875)"><path d="M1012 105l19 -16q-20 -28 -60 -62q-37 -33 -71 -49h-14q-98 35 -98 202q0 94 50 183q49 90 118 131v2q-78 13 -93 64q-2 6 -6 24t-7 25.5t-10 18.5t-18.5 15.5t-28.5 4.5q-33 0 -58 -30q-25 -28 -25 -69q9 -48 9 -116q0 -79 -10 -130q-11 -54 -45 -120&#10;q-60 -109 -147 -205q-13 8 -43 29q-35 25 -53 33q-23 -19 -45 -49l-22 14l94 145q1 -1 26.5 -19.5t45.5 -31.5t22 -11q52 0 52 323q0 159 -14 198t-61 39q-31 0 -55 -28t-24 -58v-123q0 -81 -10 -132q-9 -55 -45 -122q-59 -113 -147 -207l-36 21q-63 39 -76 41&#10;q-19 -10 -44 -49l-22 14l94 145q104 -62 110 -62q37 0 45 120q2 30 5 97q1 25 1 100v39q0 52 -8 95q-8 56 -28 87q-24 38 -63 38q-30 1 -48.5 -20t-18.5 -47t12 -47q9 -15 28.5 -31.5l35 -28.5t27.5 -35t12 -54q0 -50 -50 -94t-116 -60l-9 22q30 10 52.5 32t22.5 49&#10;q0 55 -57 94q-57 41 -57 96q0 72 53 113q54 42 132 42q55 0 104 -24.5t76 -66.5h2q18 37 55 62t77 25q99 0 142 -87h2q21 39 55.5 63t74.5 24q96 0 127 -67q5 -10 15 -40t11 -34q14 -35 51 -39v-14q-75 -35 -99 -106q-23 -72 -23 -183q0 -153 36 -153q27 2 71 50z" id="x1D510"></path></g>
<g transform="matrix(.012,-0,0,-.012,18.475,15.95)"><path d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" id="x32"></path></g><g transform="matrix(.012,-0,0,-.012,24.186,15.95)"><path d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z" id="x2C"></path></g><g transform="matrix(.012,-0,0,-.012,26.899,15.95)"><path d="M983 225q0 -112 -67 -174.5t-150 -62.5q-91 0 -154.5 43.5t-113.5 129.5q-49 -85 -104 -129t-138 -44q-98 0 -158.5 66t-60.5 154q0 59 21 106.5t54.5 75.5t70.5 43t73 15q90 0 152.5 -43.5t112.5 -128.5q48 84 104.5 128t140.5 44q93 0 155 -65t62 -158zM478 196&#10;q-27 49 -47 80t-50 67t-64 54t-73 18q-48 0 -81.5 -47t-33.5 -128q0 -96 37.5 -157.5t99.5 -61.5q68 0 117.5 47t94.5 128zM889 204q0 91 -35.5 151t-99.5 60q-68 0 -119 -47t-95 -127q27 -49 47 -80.5t50 -67.5t65 -54t74 -18q113 0 113 183z" id="x221E"></path></g>
</svg></span></td></tr><tr><td class="thead-hr" colspan="5"><hr /></td></tr><tr><td align="left">1</td><td align="center"><span style="width: 17.200001px;"><svg height="15.4" id="M96" style="vertical-align:-3.3907pt" version="1.1" viewbox="0 0 17.200001 15.4" width="17.200001" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.112)"><path d="M609 650l-19 -162l-30 -2q2 69 -14 94q-9 18 -28.5 26t-74.5 8h-88q-30 0 -37 -6.5t-12 -36.5l-41 -212h103q48 0 69 5.5t32 19t26 50.5h29l-40 -198h-30q2 58 -11.5 70.5t-85.5 12.5h-99l-30 -167q-17 -87 3 -101q18 -15 111 -15q59 0 89 7t54 27q8 7 15.5 16t16 21.5&#10;l14 20.5t15 23.5t13.5 21.5l29 -10q-52 -129 -71 -163h-500l6 28q66 4 83 17.5t28 73.5l77 409q11 61 -0.5 75.5t-78.5 18.5l10 28h467z" id="x1D438"></path></g>
<g transform="matrix(.012,-0,0,-.012,10.8,15.187)"><path d="M249 635q70 0 116 -43t46 -105q0 -46 -28 -80q-22 -25 -80 -64q62 -35 97 -75t35 -99q0 -81 -63 -131t-135 -50q-83 0 -137.5 45.5t-54.5 123.5q0 52 45 95q29 28 89 64q-109 62 -109 155q0 66 50.5 115t128.5 49zM238 603q-42 0 -67.5 -31t-25.5 -72q0 -50 32.5 -79.5&#10;t98.5 -62.5q61 48 61 124q0 59 -29.5 90t-69.5 31zM248 20q46 0 76.5 33.5t30.5 89.5q0 50 -39 85.5t-110 71.5q-81 -54 -81 -137q0 -67 35.5 -105t87.5 -38z" id="x38"></path></g>
</svg></span></td><td align="center">155520</td><td align="center">1</td><td align="center"><span style="width: 17.200001px;"><svg height="15.4" id="M97" style="vertical-align:-3.3907pt" version="1.1" viewbox="0 0 17.200001 15.4" width="17.200001" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.112)"><path d="M609 650l-19 -162l-30 -2q2 69 -14 94q-9 18 -28.5 26t-74.5 8h-88q-30 0 -37 -6.5t-12 -36.5l-41 -212h103q48 0 69 5.5t32 19t26 50.5h29l-40 -198h-30q2 58 -11.5 70.5t-85.5 12.5h-99l-30 -167q-17 -87 3 -101q18 -15 111 -15q59 0 89 7t54 27q8 7 15.5 16t16 21.5&#10;l14 20.5t15 23.5t13.5 21.5l29 -10q-52 -129 -71 -163h-500l6 28q66 4 83 17.5t28 73.5l77 409q11 61 -0.5 75.5t-78.5 18.5l10 28h467z" id="x1D438"></path></g>
<g transform="matrix(.012,-0,0,-.012,10.8,15.187)"><path d="M249 635q70 0 116 -43t46 -105q0 -46 -28 -80q-22 -25 -80 -64q62 -35 97 -75t35 -99q0 -81 -63 -131t-135 -50q-83 0 -137.5 45.5t-54.5 123.5q0 52 45 95q29 28 89 64q-109 62 -109 155q0 66 50.5 115t128.5 49zM238 603q-42 0 -67.5 -31t-25.5 -72q0 -50 32.5 -79.5&#10;t98.5 -62.5q61 48 61 124q0 59 -29.5 90t-69.5 31zM248 20q46 0 76.5 33.5t30.5 89.5q0 50 -39 85.5t-110 71.5q-81 -54 -81 -137q0 -67 35.5 -105t87.5 -38z" id="x38"></path></g>
</svg></span></td></tr><tr class="table-tr"><td colspan="5"><hr class="tbody-hr" /></td></tr></table></td></tr></table></div></div>

Table 1 | On the Classification of Lattices Over <svg      style="vertical-align:-3.06pt;width:75px;" id="M1" height="24.4125" version="1.1" viewBox="0 0 75 24.4125" width="75"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.022,-0,0,-.022,26.138,16.937)"><path id="x221A" d="M782 703l-309 -950l-231 505l-91 -47l-18 35l147 78l194 -431h2l264 810h42z" /></g><g transform="matrix(.022,-0,0,-.022,.062,20.538)"><path id="x211A" d="M794 -138l-9 -21q-74 -1 -142 15q-79 19 -254 130q-139 7 -231 87q-52 44 -83 107t-31 135q0 145 103 247.5t274 102.5q145 0 249.5 -93.5t104.5 -240.5q0 -126 -73 -212.5t-179 -116.5q137 -85 271 -140zM673 302q0 152 -78 239.5t-201 87.5q-50 0 -84.5 -74t-34.5 -209
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			<g transform="matrix(.022,-0,0,-.022,43.262,20.538)"><path id="x2212" d="M535 230h-483v50h483v-50z" /></g><g transform="matrix(.022,-0,0,-.022,56.421,20.538)"><path id="x33" d="M285 378v-2q65 -13 102 -54.5t37 -97.5q0 -57 -30.5 -104.5t-74 -75t-85.5 -42t-72 -14.5q-31 0 -59.5 11t-40.5 23q-19 18 -16 36q1 16 23 33q13 10 24 0q58 -51 124 -51q55 0 88 40t33 112q0 64 -39 96.5t-88 32.5q-29 0 -64 -11l-6 29q77 25 118 57.5t41 84.5
q0 45 -26.5 69.5t-68.5 24.5q-67 0 -120 -79l-20 20l43 63q51 56 127 56h1q66 0 107 -37t41 -95q0 -42 -31 -71q-22 -23 -68 -54z" /></g><g transform="matrix(.022,-0,0,-.022,67.181,20.538)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g>

		
	
</svg> Which Are Even Unimodular <svg     style="vertical-align:-0.0pt;width:16.875px;" id="M2" height="15.25" version="1.1" viewBox="0 0 16.875 15.25" width="16.875"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.022,-0,0,-.022,.062,15.187)"><path id="x2124" d="M734 170q-25 -130 -36 -170h-661l-8 15l420 599h-173q-77 0 -102 -7t-39 -29q-19 -29 -35 -87l-29 1l12 183h24q9 -16 19 -20.5t34 -4.5h543l7 -14l-416 -598l166 -2q94 0 131.5 7.5t58.5 29.5q28 29 55 103zM639 614h-89l-397 -567l88 -7z" /></g>

		
	
</svg>-Lattices of Rank 32

title

Table 1: On the Classification of Lattices Over <svg      style="vertical-align:-3.06pt;width:75px;" id="M1" height="24.4125" version="1.1" viewBox="0 0 75 24.4125" width="75"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.022,-0,0,-.022,26.138,16.937)"><path id="x221A" d="M782 703l-309 -950l-231 505l-91 -47l-18 35l147 78l194 -431h2l264 810h42z" /></g><g transform="matrix(.022,-0,0,-.022,.062,20.538)"><path id="x211A" d="M794 -138l-9 -21q-74 -1 -142 15q-79 19 -254 130q-139 7 -231 87q-52 44 -83 107t-31 135q0 145 103 247.5t274 102.5q145 0 249.5 -93.5t104.5 -240.5q0 -126 -73 -212.5t-179 -116.5q137 -85 271 -140zM673 302q0 152 -78 239.5t-201 87.5q-50 0 -84.5 -74t-34.5 -209
q0 -265 118 -317l66 18l31 -19q76 19 129.5 90t53.5 184zM298 54q-66 94 -66 292q0 178 50 256q-62 -32 -99 -98t-37 -162q0 -98 41 -174t111 -114z" /></g><g transform="matrix(.022,-0,0,-.022,18.377,20.538)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g>

			<g transform="matrix(1.25,0,0,-1.25,90,90)">
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</svg>-Lattices of Rank 32

description

Table 1 | On the Classification of Lattices Over 
	
		
			

			
				
			
			

		
	
 Which Are Even Unimodular 
	
		
			

		
	
-Lattices of Rank 32

Table 1

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International Journal of Mathematics and Mathematical Sciences

On the Classification of Lattices Over Which Are Even Unimodular -Lattices of Rank 32

Table 1