\(\Lambda_{\mathfrak{G}_{33}}(-1) \to U^{3} \oplus E_{8}(-1)^{2}\)
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 \(E_8(-1)^2\).
\(\Lambda_{\mathfrak{G}_{33}}(-1)(-1) = \Lambda_G \hookrightarrow \mathrm{II}_{3,19}\), row 33, complement of 02A0
A morphism \(\Lambda_{\mathfrak{G}_{33}}(-1)(-1) \to U^{3} \oplus E_{8}(-1)^{2}\): \(b_T(\varphi x, \varphi y) = -1 \, b_S(x, y)\).
The coinvariant lattice \(\Lambda_G\) of \(\mathfrak{G}_{33} \cong C_7 \rtimes C_3\) of Table 10.2 of Hashimoto, the orthogonal complement of the image of \(U(7) \oplus \begin{pmatrix} -2 & -1 \\ -1 & -4 \end{pmatrix}(-1)\). The orthogonal sum \(U(7) \oplus \begin{pmatrix} -2 & -1 \\ -1 & -4 \end{pmatrix}(-1) \oplus \Lambda_{\mathfrak{G}_{33}}(-1)(-1)\) 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.
{"matrix": [
[0, 2, 2, 3, 5, 2, -1, 0, 4, -3, 2, 1, 6, -7, 0, 3, 2, 2],
[-1, 0, 0, 1, 3, 1, -1, 0, 2, -2, 1, 0, 2, -3, 1, 2, 0, -1],
[-1, -2, -11, -2, -7, 0, -4, 3, -22, -9, -15, -4, -8, -3, 2, -14, 16, 0],
[0, 0, -1, 0, -1, 0, 0, 0, -2, -1, -1, 0, -1, 0, 0, -1, 1, 0],
[-60, 40, 55, 99, 211, 78, -72, -20, 153, -118, 52, -3, 195, -252, 19, 157, 19, 0],
[-3, 2, 3, 5, 11, 4, -4, -1, 8, -6, 3, 0, 10, -13, 1, 8, 1, 0],
[21, -15, -25, -37, -81, -30, 27, 7, -64, 40, -24, 0, -75, 94, -8, -61, -4, 0],
[31, -22, -38, -54, -120, -44, 39, 10, -96, 57, -37, 0, -111, 137, -12, -91, -4, 0],
[43, -31, -50, -76, -166, -62, 55, 13, -129, 83, -49, 0, -153, 193, -17, -124, -10, 0],
[64, -46, -75, -112, -246, -91, 81, 20, -193, 122, -74, 0, -228, 285, -25, -185, -13, 0],
[52, -37, -62, -91, -201, -75, 66, 16, -158, 99, -61, 0, -186, 232, -20, -152, -10, 0],
[41, -29, -48, -71, -157, -58, 52, 13, -123, 78, -47, 0, -145, 181, -15, -119, -8, 0],
[28, -20, -33, -49, -108, -40, 36, 9, -84, 54, -32, 0, -100, 125, -10, -82, -6, 0],
[14, -10, -18, -25, -56, -20, 18, 5, -46, 26, -18, 0, -52, 63, -5, -43, -1, 0],
[17, -11, -14, -27, -59, -21, 22, 6, -41, 35, -15, 1, -54, 72, -4, -43, -5, 0],
[25, -17, -22, -40, -89, -31, 32, 9, -64, 50, -25, -1, -81, 106, -6, -66, -6, 0],
[32, -21, -29, -50, -113, -39, 40, 11, -82, 63, -32, 0, -103, 134, -7, -84, -6, 0],
[49, -34, -45, -79, -176, -61, 62, 17, -128, 98, -50, -1, -161, 209, -11, -130, -11, 0],
[39, -27, -35, -61, -137, -47, 48, 14, -100, 76, -39, -1, -125, 163, -8, -101, -8, 0],
[28, -19, -25, -43, -98, -34, 34, 10, -71, 55, -28, -1, -89, 117, -6, -72, -6, 0],
[13, -10, -13, -21, -48, -16, 16, 5, -37, 25, -16, -1, -44, 56, -2, -36, -1, 0],
[0, -1, -1, 0, -2, 0, 0, 0, -3, 0, -3, -1, -2, 2, 1, -2, 2, 0]
],
"row_subdivisions": [],
"column_subdivisions": []}