Basic invariants
| Dimension: | $7$ |
| Group: | $C_2^3:(C_7: C_3)$ |
| Conductor: | \(100367308864\)\(\medspace = 2^{6} \cdot 199^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.100367308864.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_2^3:(C_7: C_3)$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $F_8:C_3$ |
| Projective stem field: | Galois closure of 8.0.100367308864.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} + 4x^{6} - 16x^{5} + 46x^{4} - 70x^{3} + 82x^{2} - 48x + 22 \)
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The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$:
\( x^{3} + 2x + 18 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 a^{2} + 16 a + 18 + \left(6 a^{2} + 21 a + 22\right)\cdot 23 + \left(a^{2} + 16 a + 7\right)\cdot 23^{2} + \left(4 a^{2} + a + 6\right)\cdot 23^{3} + \left(18 a^{2} + 19 a + 13\right)\cdot 23^{4} + \left(22 a^{2} + 4 a + 8\right)\cdot 23^{5} + \left(21 a^{2} + 8 a + 3\right)\cdot 23^{6} + \left(8 a^{2} + a\right)\cdot 23^{7} + \left(4 a^{2} + 17 a + 9\right)\cdot 23^{8} + \left(22 a^{2} + 15 a + 9\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 19 a^{2} + 11 a + 6 + \left(5 a^{2} + 19 a + 10\right)\cdot 23 + \left(10 a^{2} + 14 a + 22\right)\cdot 23^{2} + \left(19 a^{2} + 12 a + 3\right)\cdot 23^{3} + \left(12 a + 7\right)\cdot 23^{4} + \left(8 a^{2} + 13 a + 18\right)\cdot 23^{5} + \left(7 a^{2} + 20 a + 11\right)\cdot 23^{6} + \left(20 a^{2} + 15 a + 10\right)\cdot 23^{7} + \left(2 a + 22\right)\cdot 23^{8} + \left(a^{2} + 15 a\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 2 + 4\cdot 23 + 2\cdot 23^{2} + 16\cdot 23^{3} + 18\cdot 23^{4} + 8\cdot 23^{5} + 7\cdot 23^{6} + 19\cdot 23^{7} + 3\cdot 23^{8} +O(23^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 9 a^{2} + 15 a + 12 + \left(7 a + 7\right)\cdot 23 + \left(9 a^{2} + 2 a + 18\right)\cdot 23^{2} + \left(13 a^{2} + 18\right)\cdot 23^{3} + \left(12 a^{2} + 10 a + 5\right)\cdot 23^{4} + \left(3 a^{2} + 10 a + 21\right)\cdot 23^{5} + \left(7 a^{2} + 4 a + 21\right)\cdot 23^{6} + \left(a^{2} + 21 a + 12\right)\cdot 23^{7} + \left(9 a^{2} + 15 a + 7\right)\cdot 23^{8} + \left(15 a + 3\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 12 a^{2} + 15 a + 16 + \left(16 a^{2} + 16 a + 13\right)\cdot 23 + \left(12 a^{2} + 3 a + 15\right)\cdot 23^{2} + \left(5 a^{2} + 21 a\right)\cdot 23^{3} + \left(15 a^{2} + 16 a + 17\right)\cdot 23^{4} + \left(19 a^{2} + 7 a + 19\right)\cdot 23^{5} + \left(16 a^{2} + 10 a + 11\right)\cdot 23^{6} + \left(12 a^{2} + 20\right)\cdot 23^{7} + \left(9 a^{2} + 13 a + 15\right)\cdot 23^{8} + \left(14 a + 18\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 15 a^{2} + 9 a + 16 + \left(2 a^{2} + 11 a + 13\right)\cdot 23 + \left(19 a^{2} + 18 a + 3\right)\cdot 23^{2} + \left(18 a^{2} + 19 a + 3\right)\cdot 23^{3} + \left(16 a^{2} + 2 a + 13\right)\cdot 23^{4} + \left(21 a^{2} + 8 a + 13\right)\cdot 23^{5} + \left(5 a^{2} + 13 a + 17\right)\cdot 23^{6} + \left(22 a^{2} + 11 a + 20\right)\cdot 23^{7} + \left(20 a^{2} + 14 a + 10\right)\cdot 23^{8} + \left(14 a^{2} + 17 a + 19\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 13 + 13\cdot 23 + 21\cdot 23^{2} + 19\cdot 23^{4} + 10\cdot 23^{5} + 18\cdot 23^{6} + 19\cdot 23^{7} + 14\cdot 23^{8} + 15\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 12 a^{2} + 3 a + 12 + \left(14 a^{2} + 15 a + 6\right)\cdot 23 + \left(16 a^{2} + 12 a\right)\cdot 23^{2} + \left(7 a^{2} + 13 a + 19\right)\cdot 23^{3} + \left(5 a^{2} + 7 a + 20\right)\cdot 23^{4} + \left(16 a^{2} + a + 13\right)\cdot 23^{5} + \left(9 a^{2} + 12 a + 22\right)\cdot 23^{6} + \left(3 a^{2} + 18 a + 10\right)\cdot 23^{7} + \left(a^{2} + 5 a + 7\right)\cdot 23^{8} + \left(7 a^{2} + 13 a + 1\right)\cdot 23^{9} +O(23^{10})\)
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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 |
| $1$ | $1$ | $()$ | $7$ |
| $7$ | $2$ | $(1,4)(2,6)(3,5)(7,8)$ | $-1$ |
| $28$ | $3$ | $(1,7,2)(4,5,8)$ | $1$ |
| $28$ | $3$ | $(1,2,7)(4,8,5)$ | $1$ |
| $28$ | $6$ | $(1,5,7,8,2,4)(3,6)$ | $-1$ |
| $28$ | $6$ | $(1,4,2,8,7,5)(3,6)$ | $-1$ |
| $24$ | $7$ | $(1,6,5,4,7,8,2)$ | $0$ |
| $24$ | $7$ | $(1,4,2,5,8,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.