This is verbose output from lllgcd() in CALC.
alpha = 1
1 0 0 0 116085838
0 1 0 0 181081878
0 0 1 0 314252913
0 0 0 1 10346840
Row 2 -> Row 2 + -2 x Row 1
1 0 0 0 116085838
-2 1 0 0 -51089798
0 0 1 0 314252913
0 0 0 1 10346840
Swapping Rows 1 and 2
-2 1 0 0 -51089798
1 0 0 0 116085838
0 0 1 0 314252913
0 0 0 1 10346840
Row 2 -> Row 2 + 2 x Row 1
-2 1 0 0 -51089798
-3 2 0 0 13906242
0 0 1 0 314252913
0 0 0 1 10346840
Swapping Rows 1 and 2
-3 2 0 0 13906242
-2 1 0 0 -51089798
0 0 1 0 314252913
0 0 0 1 10346840
Row 2 -> Row 2 + 4 x Row 1
-3 2 0 0 13906242
-14 9 0 0 4535170
0 0 1 0 314252913
0 0 0 1 10346840
Swapping Rows 1 and 2
-14 9 0 0 4535170
-3 2 0 0 13906242
0 0 1 0 314252913
0 0 0 1 10346840
Row 2 -> Row 2 + -3 x Row 1
-14 9 0 0 4535170
39 -25 0 0 300732
0 0 1 0 314252913
0 0 0 1 10346840
Swapping Rows 1 and 2
39 -25 0 0 300732
-14 9 0 0 4535170
0 0 1 0 314252913
0 0 0 1 10346840
Row 2 -> Row 2 + -15 x Row 1
39 -25 0 0 300732
-599 384 0 0 24190
0 0 1 0 314252913
0 0 0 1 10346840
Swapping Rows 1 and 2
-599 384 0 0 24190
39 -25 0 0 300732
0 0 1 0 314252913
0 0 0 1 10346840
Row 2 -> Row 2 + -12 x Row 1
-599 384 0 0 24190
7227 -4633 0 0 10452
0 0 1 0 314252913
0 0 0 1 10346840
Swapping Rows 1 and 2
7227 -4633 0 0 10452
-599 384 0 0 24190
0 0 1 0 314252913
0 0 0 1 10346840
Row 2 -> Row 2 + -2 x Row 1
7227 -4633 0 0 10452
-15053 9650 0 0 3286
0 0 1 0 314252913
0 0 0 1 10346840
Swapping Rows 1 and 2
-15053 9650 0 0 3286
7227 -4633 0 0 10452
0 0 1 0 314252913
0 0 0 1 10346840
Row 2 -> Row 2 + -3 x Row 1
-15053 9650 0 0 3286
52386 -33583 0 0 594
0 0 1 0 314252913
0 0 0 1 10346840
Swapping Rows 1 and 2
52386 -33583 0 0 594
-15053 9650 0 0 3286
0 0 1 0 314252913
0 0 0 1 10346840
Row 2 -> Row 2 + -6 x Row 1
52386 -33583 0 0 594
-329369 211148 0 0 -278
0 0 1 0 314252913
0 0 0 1 10346840
Swapping Rows 1 and 2
-329369 211148 0 0 -278
52386 -33583 0 0 594
0 0 1 0 314252913
0 0 0 1 10346840
Row 2 -> Row 2 + 2 x Row 1
-329369 211148 0 0 -278
-606352 388713 0 0 38
0 0 1 0 314252913
0 0 0 1 10346840
Swapping Rows 1 and 2
-606352 388713 0 0 38
-329369 211148 0 0 -278
0 0 1 0 314252913
0 0 0 1 10346840
Row 2 -> Row 2 + 7 x Row 1
-606352 388713 0 0 38
-4573833 2932139 0 0 -12
0 0 1 0 314252913
0 0 0 1 10346840
Swapping Rows 1 and 2
-4573833 2932139 0 0 -12
-606352 388713 0 0 38
0 0 1 0 314252913
0 0 0 1 10346840
Row 2 -> Row 2 + 3 x Row 1
-4573833 2932139 0 0 -12
-14327851 9185130 0 0 2
0 0 1 0 314252913
0 0 0 1 10346840
Swapping Rows 1 and 2
-14327851 9185130 0 0 2
-4573833 2932139 0 0 -12
0 0 1 0 314252913
0 0 0 1 10346840
Row 2 -> Row 2 + 6 x Row 1
-14327851 9185130 0 0 2
-90540939 58042919 0 0 0
0 0 1 0 314252913
0 0 0 1 10346840
Swapping Rows 1 and 2
-90540939 58042919 0 0 0
-14327851 9185130 0 0 2
0 0 1 0 314252913
0 0 0 1 10346840
Row 3 -> Row 3 + -157126456 x Row 2
-90540939 58042919 0 0 0
-14327851 9185130 0 0 2
2251284449726056 -1443226924799280 1 0 1
0 0 0 1 10346840
Swapping Rows 2 and 3
-90540939 58042919 0 0 0
2251284449726056 -1443226924799280 1 0 1
-14327851 9185130 0 0 2
0 0 0 1 10346840
Row 2 -> Row 2 + 24864823 x Row 1
-90540939 58042919 0 0 0
27237259 -17460943 1 0 1
-14327851 9185130 0 0 2
0 0 0 1 10346840
Row 3 -> Row 3 + -2 x Row 2
-90540939 58042919 0 0 0
27237259 -17460943 1 0 1
-68802369 44107016 -2 0 0
0 0 0 1 10346840
Swapping Rows 2 and 3
-90540939 58042919 0 0 0
-68802369 44107016 -2 0 0
27237259 -17460943 1 0 1
0 0 0 1 10346840
Row 2 -> Row 2 + -1 x Row 1
-90540939 58042919 0 0 0
21738570 -13935903 -2 0 0
27237259 -17460943 1 0 1
0 0 0 1 10346840
Swapping Rows 1 and 2
21738570 -13935903 -2 0 0
-90540939 58042919 0 0 0
27237259 -17460943 1 0 1
0 0 0 1 10346840
Row 2 -> Row 2 + 4 x Row 1
21738570 -13935903 -2 0 0
-3586659 2299307 -8 0 0
27237259 -17460943 1 0 1
0 0 0 1 10346840
Swapping Rows 1 and 2
-3586659 2299307 -8 0 0
21738570 -13935903 -2 0 0
27237259 -17460943 1 0 1
0 0 0 1 10346840
Row 2 -> Row 2 + 6 x Row 1
-3586659 2299307 -8 0 0
218616 -140061 -50 0 0
27237259 -17460943 1 0 1
0 0 0 1 10346840
Swapping Rows 1 and 2
218616 -140061 -50 0 0
-3586659 2299307 -8 0 0
27237259 -17460943 1 0 1
0 0 0 1 10346840
Row 2 -> Row 2 + 16 x Row 1
218616 -140061 -50 0 0
-88803 58331 -808 0 0
27237259 -17460943 1 0 1
0 0 0 1 10346840
Swapping Rows 1 and 2
-88803 58331 -808 0 0
218616 -140061 -50 0 0
27237259 -17460943 1 0 1
0 0 0 1 10346840
Row 2 -> Row 2 + 2 x Row 1
-88803 58331 -808 0 0
41010 -23399 -1666 0 0
27237259 -17460943 1 0 1
0 0 0 1 10346840
Swapping Rows 1 and 2
41010 -23399 -1666 0 0
-88803 58331 -808 0 0
27237259 -17460943 1 0 1
0 0 0 1 10346840
Row 2 -> Row 2 + 2 x Row 1
41010 -23399 -1666 0 0
-6783 11533 -4140 0 0
27237259 -17460943 1 0 1
0 0 0 1 10346840
Swapping Rows 1 and 2
-6783 11533 -4140 0 0
41010 -23399 -1666 0 0
27237259 -17460943 1 0 1
0 0 0 1 10346840
Row 2 -> Row 2 + 3 x Row 1
-6783 11533 -4140 0 0
20661 11200 -14086 0 0
27237259 -17460943 1 0 1
0 0 0 1 10346840
Row 3 -> Row 3 + -623 x Row 2
-6783 11533 -4140 0 0
20661 11200 -14086 0 0
14365456 -24438543 8775579 0 1
0 0 0 1 10346840
Row 3 -> Row 3 + 2119 x Row 1
-6783 11533 -4140 0 0
20661 11200 -14086 0 0
-7721 -116 2919 0 1
0 0 0 1 10346840
Row 4 -> Row 4 + -10346840 x Row 3
-6783 11533 -4140 0 0
20661 11200 -14086 0 0
-7721 -116 2919 0 1
79887951640 1200233440 -30202425960 1 0
Swapping Rows 3 and 4
-6783 11533 -4140 0 0
20661 11200 -14086 0 0
79887951640 1200233440 -30202425960 1 0
-7721 -116 2919 0 1
Row 3 -> Row 3 + -2957773 x Row 2
-6783 11533 -4140 0 0
20661 11200 -14086 0 0
18777403687 -31926824160 11460764518 1 0
-7721 -116 2919 0 1
Swapping Rows 2 and 3
-6783 11533 -4140 0 0
18777403687 -31926824160 11460764518 1 0
20661 11200 -14086 0 0
-7721 -116 2919 0 1
Row 2 -> Row 2 + 2768302 x Row 1
-6783 11533 -4140 0 0
11221 2806 -5762 1 0
20661 11200 -14086 0 0
-7721 -116 2919 0 1
Swapping Rows 1 and 2
11221 2806 -5762 1 0
-6783 11533 -4140 0 0
20661 11200 -14086 0 0
-7721 -116 2919 0 1
Swapping Rows 2 and 3
11221 2806 -5762 1 0
20661 11200 -14086 0 0
-6783 11533 -4140 0 0
-7721 -116 2919 0 1
Row 2 -> Row 2 + -2 x Row 1
11221 2806 -5762 1 0
-1781 5588 -2562 -2 0
-6783 11533 -4140 0 0
-7721 -116 2919 0 1
Swapping Rows 1 and 2
-1781 5588 -2562 -2 0
11221 2806 -5762 1 0
-6783 11533 -4140 0 0
-7721 -116 2919 0 1
Swapping Rows 2 and 3
-1781 5588 -2562 -2 0
-6783 11533 -4140 0 0
11221 2806 -5762 1 0
-7721 -116 2919 0 1
Row 2 -> Row 2 + -2 x Row 1
-1781 5588 -2562 -2 0
-3221 357 984 4 0
11221 2806 -5762 1 0
-7721 -116 2919 0 1
Swapping Rows 1 and 2
-3221 357 984 4 0
-1781 5588 -2562 -2 0
11221 2806 -5762 1 0
-7721 -116 2919 0 1
Row 3 -> Row 3 + -1 x Row 2
-3221 357 984 4 0
-1781 5588 -2562 -2 0
13002 -2782 -3200 3 0
-7721 -116 2919 0 1
Swapping Rows 2 and 3
-3221 357 984 4 0
13002 -2782 -3200 3 0
-1781 5588 -2562 -2 0
-7721 -116 2919 0 1
Row 2 -> Row 2 + 4 x Row 1
-3221 357 984 4 0
118 -1354 736 19 0
-1781 5588 -2562 -2 0
-7721 -116 2919 0 1
Swapping Rows 1 and 2
118 -1354 736 19 0
-3221 357 984 4 0
-1781 5588 -2562 -2 0
-7721 -116 2919 0 1
Swapping Rows 2 and 3
118 -1354 736 19 0
-1781 5588 -2562 -2 0
-3221 357 984 4 0
-7721 -116 2919 0 1
Row 2 -> Row 2 + 4 x Row 1
118 -1354 736 19 0
-1309 172 382 74 0
-3221 357 984 4 0
-7721 -116 2919 0 1
Swapping Rows 1 and 2
-1309 172 382 74 0
118 -1354 736 19 0
-3221 357 984 4 0
-7721 -116 2919 0 1
Swapping Rows 2 and 3
-1309 172 382 74 0
-3221 357 984 4 0
118 -1354 736 19 0
-7721 -116 2919 0 1
Row 2 -> Row 2 + -2 x Row 1
-1309 172 382 74 0
-603 13 220 -144 0
118 -1354 736 19 0
-7721 -116 2919 0 1
Swapping Rows 1 and 2
-603 13 220 -144 0
-1309 172 382 74 0
118 -1354 736 19 0
-7721 -116 2919 0 1
Row 2 -> Row 2 + -2 x Row 1
-603 13 220 -144 0
-103 146 -58 362 0
118 -1354 736 19 0
-7721 -116 2919 0 1
Swapping Rows 1 and 2
-103 146 -58 362 0
-603 13 220 -144 0
118 -1354 736 19 0
-7721 -116 2919 0 1
Row 3 -> Row 3 + 1 x Row 1
-103 146 -58 362 0
-603 13 220 -144 0
15 -1208 678 381 0
-7721 -116 2919 0 1
Row 4 -> Row 4 + -1 x Row 3
-103 146 -58 362 0
-603 13 220 -144 0
15 -1208 678 381 0
-7736 1092 2241 -381 1
Row 4 -> Row 4 + -12 x Row 2
-103 146 -58 362 0
-603 13 220 -144 0
15 -1208 678 381 0
-500 936 -399 1347 1
Row 4 -> Row 4 + -4 x Row 1
-103 146 -58 362 0
-603 13 220 -144 0
15 -1208 678 381 0
-88 352 -167 -101 1
The unimodular matrix found by LLL is
-103 146 -58 362
-603 13 220 -144
15 -1208 678 381
-88 352 -167 -101
gcd(116085838,181081878,314252913,10346840)=1
The multipliers found by LLL are (-88,352,-167,-101)
The Euclidean norm squared = 169738
These are the smallest multipliers.
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