Basic invariants
| Dimension: | $12$ |
| Group: | $(A_4\wr C_2):C_2$ |
| Conductor: | \(507\!\cdots\!824\)\(\medspace = 2^{28} \cdot 3^{6} \cdot 11^{10} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.4.642318336.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 24T1503 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $\PGOPlus(4,3)$ |
| Projective stem field: | Galois closure of 8.4.642318336.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{6} - 4x^{5} + 6x^{4} + 8x^{3} - 4x - 2 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$:
\( x^{3} + 2x + 11 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 a^{2} + 4 a + 5 + \left(10 a^{2} + 3 a + 4\right)\cdot 13 + \left(10 a^{2} + 12 a + 10\right)\cdot 13^{2} + \left(7 a^{2} + 8 a + 1\right)\cdot 13^{3} + \left(8 a^{2} + 7 a + 8\right)\cdot 13^{4} + \left(9 a^{2} + 9 a + 8\right)\cdot 13^{5} + \left(4 a^{2} + 2 a + 11\right)\cdot 13^{6} + \left(10 a^{2} + 10 a + 11\right)\cdot 13^{7} + \left(9 a^{2} + 6 a + 10\right)\cdot 13^{8} + \left(11 a^{2} + 9 a + 4\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 4 + 10\cdot 13 + 4\cdot 13^{2} + 9\cdot 13^{3} + 12\cdot 13^{5} + 2\cdot 13^{6} + 11\cdot 13^{7} + 8\cdot 13^{8} + 10\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 7 a^{2} + 9 a + 3 + \left(10 a^{2} + 7 a + 9\right)\cdot 13 + \left(5 a^{2} + 3 a + 3\right)\cdot 13^{2} + \left(6 a^{2} + 6 a + 4\right)\cdot 13^{3} + \left(4 a^{2} + 4 a + 11\right)\cdot 13^{4} + \left(7 a^{2} + 7 a + 9\right)\cdot 13^{5} + \left(10 a^{2} + 8 a + 10\right)\cdot 13^{6} + \left(10 a^{2} + 7 a + 3\right)\cdot 13^{7} + \left(11 a^{2} + 10 a + 9\right)\cdot 13^{8} + \left(7 a^{2} + 6 a + 12\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 6 + 13 + 12\cdot 13^{2} + 12\cdot 13^{3} + 9\cdot 13^{4} + 12\cdot 13^{5} + 9\cdot 13^{6} + 5\cdot 13^{7} + 6\cdot 13^{8} + 6\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 11 a^{2} + 6 a + 9 + \left(a^{2} + 8 a + 7\right)\cdot 13 + \left(4 a^{2} + 2 a + 12\right)\cdot 13^{2} + \left(8 a^{2} + 10 a + 7\right)\cdot 13^{3} + \left(12 a^{2} + a + 3\right)\cdot 13^{4} + \left(5 a + 10\right)\cdot 13^{5} + \left(5 a^{2} + 4 a + 5\right)\cdot 13^{6} + \left(11 a^{2} + 3 a + 11\right)\cdot 13^{7} + \left(8 a^{2} + 3 a + 8\right)\cdot 13^{8} + \left(10 a^{2} + 10\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 4 a^{2} + 12 + \left(5 a^{2} + 2 a + 10\right)\cdot 13 + \left(9 a^{2} + 10 a + 12\right)\cdot 13^{2} + \left(11 a^{2} + 10 a + 6\right)\cdot 13^{3} + \left(12 a^{2} + 9\right)\cdot 13^{4} + \left(8 a^{2} + 9 a + 7\right)\cdot 13^{5} + \left(10 a^{2} + a + 6\right)\cdot 13^{6} + \left(4 a^{2} + 8 a + 4\right)\cdot 13^{7} + \left(4 a^{2} + 8 a + 12\right)\cdot 13^{8} + \left(6 a^{2} + 9 a + 1\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 8 a^{2} + 2 a + 5 + \left(11 a + 10\right)\cdot 13 + \left(a^{2} + 7 a + 12\right)\cdot 13^{2} + \left(10 a^{2} + 4 a + 5\right)\cdot 13^{3} + \left(4 a^{2} + 11 a + 10\right)\cdot 13^{4} + 7 a\cdot 13^{5} + \left(3 a^{2} + 9 a + 3\right)\cdot 13^{6} + \left(a^{2} + 2\right)\cdot 13^{7} + \left(a^{2} + 3 a + 7\right)\cdot 13^{8} + \left(11 a^{2} + a + 2\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 7 a^{2} + 5 a + 8 + \left(10 a^{2} + 6 a + 10\right)\cdot 13 + \left(7 a^{2} + 2 a + 8\right)\cdot 13^{2} + \left(7 a^{2} + 11 a + 2\right)\cdot 13^{3} + \left(8 a^{2} + 12 a + 11\right)\cdot 13^{4} + \left(11 a^{2} + 12 a + 2\right)\cdot 13^{5} + \left(4 a^{2} + 11 a + 1\right)\cdot 13^{6} + \left(8 a + 1\right)\cdot 13^{7} + \left(3 a^{2} + 6 a + 1\right)\cdot 13^{8} + \left(4 a^{2} + 11 a + 2\right)\cdot 13^{9} +O(13^{10})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 8 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 8 }$ | Character value |
| $1$ | $1$ | $()$ | $12$ |
| $6$ | $2$ | $(1,4)(3,6)$ | $4$ |
| $9$ | $2$ | $(1,4)(2,7)(3,6)(5,8)$ | $-4$ |
| $12$ | $2$ | $(1,2)(3,5)(4,7)(6,8)$ | $0$ |
| $12$ | $2$ | $(1,8)(2,3)(4,5)(6,7)$ | $0$ |
| $36$ | $2$ | $(4,6)(7,8)$ | $0$ |
| $16$ | $3$ | $(1,6,3)$ | $-3$ |
| $32$ | $3$ | $(1,4,6)(2,7,8)$ | $0$ |
| $32$ | $3$ | $(1,4,6)(5,7,8)$ | $0$ |
| $36$ | $4$ | $(1,7,4,2)(3,8,6,5)$ | $0$ |
| $36$ | $4$ | $(1,5,4,8)(2,3,7,6)$ | $0$ |
| $36$ | $4$ | $(1,4,3,6)(2,8,5,7)$ | $0$ |
| $72$ | $4$ | $(2,8,5,7)(4,6)$ | $0$ |
| $48$ | $6$ | $(1,6,3)(2,7)(5,8)$ | $1$ |
| $96$ | $6$ | $(1,7,4,8,6,2)(3,5)$ | $0$ |
| $96$ | $6$ | $(1,5,4,7,6,8)(2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.