\(\Lambda_{\mathfrak{G}_{70}}(-1) \to U^{3} \oplus E_{8}(-1)^{2}\)
Column \(j\) of each matrix gives \(\varphi(e_j)\) in the chosen basis of the target; the subdivisions separate orthogonal summands of the source (columns) and of the target (rows).
Notes
The invariant and coinvariant lattices of the finite symplectic
groups of Hashimoto in the K3 lattice. The anti-isometry maps the
generators of the normal form of SageMath of one discriminant form to
those of the other twisted by \(-1\);
three hyperbolic planes are split off and qfisom of PARI/GP
maps the remainder onto \(E_8(-1)^2\).
\(\Lambda_{\mathfrak{G}_{70}}(-1)(-1) = \Lambda_G \hookrightarrow \mathrm{II}_{3,19}\), row 70, complement of 02AE
A morphism \(\Lambda_{\mathfrak{G}_{70}}(-1)(-1) \to U^{3} \oplus E_{8}(-1)^{2}\): \(b_T(\varphi x, \varphi y) = -1 \, b_S(x, y)\).
The coinvariant lattice \(\Lambda_G\) of \(\mathfrak{G}_{70} \cong \mathfrak{S}_5\) of Table 10.2 of Hashimoto, the orthogonal complement of the image of \(\begin{pmatrix} 4 & 1 \\ 1 & 4 \end{pmatrix} \oplus \langle 20 \rangle\). The orthogonal sum \(\begin{pmatrix} 4 & 1 \\ 1 & 4 \end{pmatrix} \oplus \langle 20 \rangle \oplus \Lambda_{\mathfrak{G}_{70}}(-1)(-1)\) glued along an anti-isometry of the discriminant forms is even, unimodular and of signature \((3, 19)\); its isometry onto the basis of \(\mathrm{II}_{3,19}\) restricts to this map.
\[\left(\begin{array}{rrrrrrrrrrrrrrrrrrr}
-2 & -1 & 0 & -1 & -2 & -1 & 0 & 1 & 0 & -1 & 1 & -1 & 1 & 1 & 1 & 1 & 0 & -2 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 \\
-16 & -11 & 1 & -10 & -15 & -6 & 0 & 11 & -5 & -8 & 9 & -7 & 23 & 12 & 8 & 10 & -1 & 0 & 12 \\
-4 & -3 & 0 & -2 & -4 & -2 & -1 & 2 & -1 & -2 & 2 & -2 & 6 & 3 & 2 & 2 & -1 & 0 & 3 \\
19 & 17 & -6 & 17 & 13 & 14 & 10 & -14 & 9 & 18 & 0 & 6 & -27 & -14 & -3 & -26 & -3 & 0 & -12 \\
1 & 1 & 0 & 1 & 1 & 1 & 1 & -1 & 0 & 1 & 0 & 0 & -2 & -1 & 0 & -2 & 0 & 0 & -1 \\
-2 & 0 & -1 & 1 & -2 & 1 & 2 & 1 & 0 & -1 & 2 & -2 & 1 & 0 & 3 & -3 & -1 & 0 & 1 \\
-5 & -2 & -2 & 1 & -6 & 1 & 3 & 2 & 0 & -1 & 5 & -5 & 6 & 1 & 6 & -4 & -2 & 0 & 3 \\
-6 & -3 & -2 & 1 & -7 & 0 & 2 & 3 & 0 & -3 & 4 & -6 & 7 & 2 & 7 & -2 & -2 & 0 & 4 \\
-10 & -5 & -3 & 0 & -11 & 0 & 3 & 5 & 0 & -5 & 7 & -9 & 13 & 3 & 10 & -3 & -3 & 0 & 7 \\
-9 & -5 & -2 & -1 & -9 & -1 & 2 & 5 & 0 & -5 & 5 & -7 & 11 & 3 & 8 & -1 & -2 & 0 & 6 \\
-7 & -4 & -2 & -1 & -7 & -1 & 1 & 4 & 0 & -4 & 4 & -5 & 9 & 2 & 6 & 0 & -1 & 0 & 5 \\
-5 & -3 & -1 & -1 & -5 & -1 & 0 & 3 & 0 & -4 & 2 & -3 & 7 & 2 & 4 & 1 & -1 & 0 & 3 \\
1 & 1 & -1 & 2 & 0 & 2 & 1 & -1 & 1 & 0 & 0 & -1 & -1 & -1 & 1 & -3 & -1 & 0 & -1 \\
11 & 9 & -1 & 7 & 10 & 6 & 2 & -7 & 3 & 6 & -4 & 5 & -16 & -8 & -4 & -9 & 2 & 0 & -8 \\
17 & 13 & -1 & 11 & 14 & 10 & 3 & -11 & 5 & 9 & -8 & 7 & -24 & -12 & -6 & -13 & 2 & 0 & -12 \\
21 & 17 & -1 & 13 & 18 & 12 & 4 & -14 & 6 & 11 & -9 & 9 & -31 & -15 & -7 & -16 & 3 & 0 & -15 \\
31 & 25 & -2 & 20 & 26 & 18 & 6 & -20 & 9 & 16 & -14 & 13 & -45 & -22 & -11 & -24 & 4 & 0 & -23 \\
26 & 20 & -1 & 17 & 22 & 14 & 5 & -17 & 8 & 13 & -12 & 11 & -38 & -18 & -10 & -20 & 3 & 0 & -19 \\
20 & 15 & -1 & 12 & 17 & 10 & 2 & -12 & 6 & 9 & -10 & 9 & -28 & -14 & -9 & -12 & 3 & 0 & -15 \\
14 & 11 & -1 & 9 & 12 & 7 & 2 & -9 & 4 & 7 & -6 & 6 & -20 & -10 & -6 & -9 & 2 & 0 & -11 \\
0 & 0 & 0 & 0 & -1 & 0 & -1 & 0 & 0 & -1 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 \\
\end{array}\right)\]
{"matrix": [
[-2, -1, 0, -1, -2, -1, 0, 1, 0, -1, 1, -1, 1, 1, 1, 1, 0, -2, 1],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0],
[-16, -11, 1, -10, -15, -6, 0, 11, -5, -8, 9, -7, 23, 12, 8, 10, -1, 0, 12],
[-4, -3, 0, -2, -4, -2, -1, 2, -1, -2, 2, -2, 6, 3, 2, 2, -1, 0, 3],
[19, 17, -6, 17, 13, 14, 10, -14, 9, 18, 0, 6, -27, -14, -3, -26, -3, 0, -12],
[1, 1, 0, 1, 1, 1, 1, -1, 0, 1, 0, 0, -2, -1, 0, -2, 0, 0, -1],
[-2, 0, -1, 1, -2, 1, 2, 1, 0, -1, 2, -2, 1, 0, 3, -3, -1, 0, 1],
[-5, -2, -2, 1, -6, 1, 3, 2, 0, -1, 5, -5, 6, 1, 6, -4, -2, 0, 3],
[-6, -3, -2, 1, -7, 0, 2, 3, 0, -3, 4, -6, 7, 2, 7, -2, -2, 0, 4],
[-10, -5, -3, 0, -11, 0, 3, 5, 0, -5, 7, -9, 13, 3, 10, -3, -3, 0, 7],
[-9, -5, -2, -1, -9, -1, 2, 5, 0, -5, 5, -7, 11, 3, 8, -1, -2, 0, 6],
[-7, -4, -2, -1, -7, -1, 1, 4, 0, -4, 4, -5, 9, 2, 6, 0, -1, 0, 5],
[-5, -3, -1, -1, -5, -1, 0, 3, 0, -4, 2, -3, 7, 2, 4, 1, -1, 0, 3],
[1, 1, -1, 2, 0, 2, 1, -1, 1, 0, 0, -1, -1, -1, 1, -3, -1, 0, -1],
[11, 9, -1, 7, 10, 6, 2, -7, 3, 6, -4, 5, -16, -8, -4, -9, 2, 0, -8],
[17, 13, -1, 11, 14, 10, 3, -11, 5, 9, -8, 7, -24, -12, -6, -13, 2, 0, -12],
[21, 17, -1, 13, 18, 12, 4, -14, 6, 11, -9, 9, -31, -15, -7, -16, 3, 0, -15],
[31, 25, -2, 20, 26, 18, 6, -20, 9, 16, -14, 13, -45, -22, -11, -24, 4, 0, -23],
[26, 20, -1, 17, 22, 14, 5, -17, 8, 13, -12, 11, -38, -18, -10, -20, 3, 0, -19],
[20, 15, -1, 12, 17, 10, 2, -12, 6, 9, -10, 9, -28, -14, -9, -12, 3, 0, -15],
[14, 11, -1, 9, 12, 7, 2, -9, 4, 7, -6, 6, -20, -10, -6, -9, 2, 0, -11],
[0, 0, 0, 0, -1, 0, -1, 0, 0, -1, 0, 0, 1, 1, 0, 1, 0, 0, 0]
],
"row_subdivisions": [],
"column_subdivisions": []}
\(\Lambda_{\mathfrak{G}_{70}}(-1)(-1) = \Lambda_G \hookrightarrow \mathrm{II}_{3,19}\), row 70, complement of 02AF
A morphism \(\Lambda_{\mathfrak{G}_{70}}(-1)(-1) \to U^{3} \oplus E_{8}(-1)^{2}\): \(b_T(\varphi x, \varphi y) = -1 \, b_S(x, y)\).
The coinvariant lattice \(\Lambda_G\) of \(\mathfrak{G}_{70} \cong \mathfrak{S}_5\) of Table 10.2 of Hashimoto, the orthogonal complement of the image of \(\begin{pmatrix} 4 & 2 & 2 \\ 2 & 6 & 1 \\ 2 & 1 & 16 \end{pmatrix}\). The orthogonal sum \(\begin{pmatrix} 4 & 2 & 2 \\ 2 & 6 & 1 \\ 2 & 1 & 16 \end{pmatrix} \oplus \Lambda_{\mathfrak{G}_{70}}(-1)(-1)\) glued along an anti-isometry of the discriminant forms is even, unimodular and of signature \((3, 19)\); its isometry onto the basis of \(\mathrm{II}_{3,19}\) restricts to this map.
\[\left(\begin{array}{rrrrrrrrrrrrrrrrrrr}
1 & 2 & -1 & -1 & 3 & -3 & -4 & -2 & 3 & 0 & -2 & 4 & -2 & -3 & -5 & 0 & 0 & -3 & -2 \\
0 & 0 & 0 & -1 & 1 & -2 & -2 & -1 & 2 & 0 & -1 & 2 & -1 & -1 & -2 & 0 & 0 & -2 & 1 \\
-101 & -60 & 70 & 88 & -125 & 172 & 243 & 108 & -241 & 2 & 73 & -192 & 97 & 96 & 365 & -62 & 79 & 203 & 0 \\
-3 & -2 & 2 & 3 & -4 & 5 & 7 & 3 & -7 & 0 & 2 & -6 & 3 & 3 & 11 & -2 & 2 & 6 & 0 \\
-284 & -178 & 193 & 257 & -368 & 481 & 669 & 289 & -668 & -1 & 202 & -556 & 295 & 276 & 1018 & -191 & 204 & 564 & 0 \\
-18 & -11 & 12 & 16 & -23 & 30 & 42 & 18 & -42 & 0 & 13 & -35 & 18 & 17 & 64 & -12 & 13 & 35 & 0 \\
-152 & -94 & 102 & 137 & -196 & 255 & 356 & 153 & -356 & 1 & 109 & -296 & 155 & 145 & 543 & -101 & 110 & 299 & 0 \\
-225 & -139 & 151 & 203 & -289 & 377 & 527 & 226 & -526 & 1 & 161 & -437 & 229 & 214 & 804 & -151 & 163 & 443 & 0 \\
-301 & -186 & 202 & 272 & -388 & 506 & 706 & 303 & -705 & 2 & 216 & -586 & 307 & 287 & 1077 & -202 & 218 & 593 & 0 \\
-451 & -279 & 303 & 408 & -581 & 759 & 1059 & 454 & -1057 & 3 & 323 & -879 & 460 & 430 & 1615 & -303 & 327 & 890 & 0 \\
-382 & -236 & 257 & 345 & -491 & 642 & 896 & 384 & -895 & 2 & 273 & -744 & 389 & 364 & 1367 & -257 & 277 & 753 & 0 \\
-311 & -192 & 209 & 280 & -399 & 521 & 728 & 313 & -727 & 1 & 221 & -604 & 315 & 296 & 1111 & -208 & 225 & 611 & 0 \\
-214 & -132 & 144 & 193 & -275 & 359 & 501 & 215 & -501 & 1 & 152 & -416 & 217 & 204 & 766 & -143 & 155 & 421 & 0 \\
-111 & -69 & 75 & 101 & -143 & 187 & 261 & 112 & -261 & 1 & 79 & -217 & 113 & 106 & 399 & -74 & 81 & 220 & 0 \\
-56 & -35 & 39 & 52 & -74 & 98 & 136 & 58 & -135 & -1 & 42 & -113 & 58 & 57 & 205 & -38 & 39 & 115 & 0 \\
-85 & -54 & 60 & 78 & -112 & 147 & 205 & 87 & -204 & -2 & 63 & -171 & 87 & 87 & 309 & -57 & 59 & 172 & 0 \\
-106 & -67 & 74 & 99 & -141 & 186 & 258 & 110 & -256 & -2 & 80 & -215 & 110 & 109 & 389 & -71 & 74 & 217 & 0 \\
-176 & -111 & 123 & 162 & -232 & 305 & 425 & 181 & -422 & -4 & 131 & -354 & 181 & 179 & 640 & -118 & 123 & 356 & 0 \\
-152 & -96 & 106 & 139 & -200 & 262 & 366 & 156 & -363 & -3 & 113 & -305 & 156 & 154 & 551 & -101 & 106 & 306 & 0 \\
-122 & -77 & 85 & 111 & -161 & 211 & 293 & 125 & -291 & -3 & 90 & -245 & 126 & 123 & 442 & -81 & 85 & 245 & 0 \\
-71 & -45 & 50 & 64 & -93 & 122 & 170 & 72 & -169 & -2 & 52 & -142 & 73 & 71 & 256 & -47 & 49 & 142 & 0 \\
-14 & -9 & 11 & 13 & -19 & 25 & 35 & 14 & -34 & -1 & 10 & -29 & 14 & 15 & 52 & -9 & 9 & 29 & 0 \\
\end{array}\right)\]
{"matrix": [
[1, 2, -1, -1, 3, -3, -4, -2, 3, 0, -2, 4, -2, -3, -5, 0, 0, -3, -2],
[0, 0, 0, -1, 1, -2, -2, -1, 2, 0, -1, 2, -1, -1, -2, 0, 0, -2, 1],
[-101, -60, 70, 88, -125, 172, 243, 108, -241, 2, 73, -192, 97, 96, 365, -62, 79, 203, 0],
[-3, -2, 2, 3, -4, 5, 7, 3, -7, 0, 2, -6, 3, 3, 11, -2, 2, 6, 0],
[-284, -178, 193, 257, -368, 481, 669, 289, -668, -1, 202, -556, 295, 276, 1018, -191, 204, 564, 0],
[-18, -11, 12, 16, -23, 30, 42, 18, -42, 0, 13, -35, 18, 17, 64, -12, 13, 35, 0],
[-152, -94, 102, 137, -196, 255, 356, 153, -356, 1, 109, -296, 155, 145, 543, -101, 110, 299, 0],
[-225, -139, 151, 203, -289, 377, 527, 226, -526, 1, 161, -437, 229, 214, 804, -151, 163, 443, 0],
[-301, -186, 202, 272, -388, 506, 706, 303, -705, 2, 216, -586, 307, 287, 1077, -202, 218, 593, 0],
[-451, -279, 303, 408, -581, 759, 1059, 454, -1057, 3, 323, -879, 460, 430, 1615, -303, 327, 890, 0],
[-382, -236, 257, 345, -491, 642, 896, 384, -895, 2, 273, -744, 389, 364, 1367, -257, 277, 753, 0],
[-311, -192, 209, 280, -399, 521, 728, 313, -727, 1, 221, -604, 315, 296, 1111, -208, 225, 611, 0],
[-214, -132, 144, 193, -275, 359, 501, 215, -501, 1, 152, -416, 217, 204, 766, -143, 155, 421, 0],
[-111, -69, 75, 101, -143, 187, 261, 112, -261, 1, 79, -217, 113, 106, 399, -74, 81, 220, 0],
[-56, -35, 39, 52, -74, 98, 136, 58, -135, -1, 42, -113, 58, 57, 205, -38, 39, 115, 0],
[-85, -54, 60, 78, -112, 147, 205, 87, -204, -2, 63, -171, 87, 87, 309, -57, 59, 172, 0],
[-106, -67, 74, 99, -141, 186, 258, 110, -256, -2, 80, -215, 110, 109, 389, -71, 74, 217, 0],
[-176, -111, 123, 162, -232, 305, 425, 181, -422, -4, 131, -354, 181, 179, 640, -118, 123, 356, 0],
[-152, -96, 106, 139, -200, 262, 366, 156, -363, -3, 113, -305, 156, 154, 551, -101, 106, 306, 0],
[-122, -77, 85, 111, -161, 211, 293, 125, -291, -3, 90, -245, 126, 123, 442, -81, 85, 245, 0],
[-71, -45, 50, 64, -93, 122, 170, 72, -169, -2, 52, -142, 73, 71, 256, -47, 49, 142, 0],
[-14, -9, 11, 13, -19, 25, 35, 14, -34, -1, 10, -29, 14, 15, 52, -9, 9, 29, 0]
],
"row_subdivisions": [],
"column_subdivisions": []}