Basic invariants
| Dimension: | $7$ |
| Group: | $C_2^3:(C_7: C_3)$ |
| Conductor: | \(30489216744448\)\(\medspace = 2^{10} \cdot 7^{5} \cdot 11^{6} \) |
| Artin stem field: | Galois closure of 8.0.4355602392064.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | 24T283 |
| Parity: | even |
| Determinant: | 1.7.3t1.a.a |
| Projective image: | $F_8:C_3$ |
| Projective stem field: | Galois closure of 8.0.4355602392064.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 12x^{6} - 20x^{5} + 46x^{4} - 94x^{3} + 126x^{2} - 110x + 49 \)
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The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$:
\( x^{3} + x + 14 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 a^{2} + 12 a + 13 + 8 a^{2} 17 + \left(a^{2} + a + 3\right)\cdot 17^{2} + \left(a^{2} + 9 a\right)\cdot 17^{3} + \left(6 a^{2} + 11 a + 4\right)\cdot 17^{4} + \left(9 a^{2} + 13 a + 8\right)\cdot 17^{5} + \left(a^{2} + 6 a + 8\right)\cdot 17^{6} + \left(3 a^{2} + 11 a + 2\right)\cdot 17^{7} + \left(6 a^{2} + 14 a + 14\right)\cdot 17^{8} + \left(6 a^{2} + 8 a + 12\right)\cdot 17^{9} +O(17^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 15 + 6\cdot 17^{2} + 2\cdot 17^{3} + 10\cdot 17^{4} + 16\cdot 17^{5} + 15\cdot 17^{6} + 15\cdot 17^{7} + 16\cdot 17^{8} + 13\cdot 17^{9} +O(17^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 7 a^{2} + 14 a + \left(8 a^{2} + a\right)\cdot 17 + \left(13 a^{2} + 15 a + 14\right)\cdot 17^{2} + \left(15 a^{2} + 16 a + 3\right)\cdot 17^{3} + \left(16 a^{2} + 6 a + 12\right)\cdot 17^{4} + \left(14 a^{2} + 11 a + 4\right)\cdot 17^{5} + \left(4 a^{2} + 8 a + 7\right)\cdot 17^{6} + \left(a^{2} + 7 a + 16\right)\cdot 17^{7} + \left(14 a^{2} + 5 a + 7\right)\cdot 17^{8} + \left(12 a^{2} + 15 a + 7\right)\cdot 17^{9} +O(17^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 4 + 13\cdot 17 + 6\cdot 17^{2} + 2\cdot 17^{3} + 4\cdot 17^{4} + 10\cdot 17^{5} + 4\cdot 17^{7} + 8\cdot 17^{8} + 14\cdot 17^{9} +O(17^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 7 a^{2} + 15 a + \left(12 a^{2} + 10 a + 14\right)\cdot 17 + \left(3 a^{2} + 4 a + 1\right)\cdot 17^{2} + \left(a^{2} + 9 a + 11\right)\cdot 17^{3} + \left(2 a + 6\right)\cdot 17^{4} + \left(16 a^{2} + 6 a + 5\right)\cdot 17^{5} + \left(5 a^{2} + 12 a + 2\right)\cdot 17^{6} + \left(14 a^{2} + 7 a + 8\right)\cdot 17^{7} + \left(7 a^{2} + 6 a + 9\right)\cdot 17^{8} + \left(7 a^{2} + 7 a + 9\right)\cdot 17^{9} +O(17^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( a^{2} + 10 a + 6 + \left(7 a^{2} + 9 a + 11\right)\cdot 17 + \left(5 a^{2} + 16 a + 5\right)\cdot 17^{2} + \left(6 a^{2} + 8 a + 9\right)\cdot 17^{3} + \left(11 a^{2} + 7 a + 7\right)\cdot 17^{4} + \left(2 a^{2} + 5 a + 9\right)\cdot 17^{5} + \left(10 a^{2} + a + 8\right)\cdot 17^{6} + \left(12 a^{2} + 5 a + 14\right)\cdot 17^{7} + \left(13 a^{2} + 15 a + 7\right)\cdot 17^{8} + \left(14 a^{2} + 4 a + 1\right)\cdot 17^{9} +O(17^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 3 a^{2} + 5 a + 3 + \left(13 a^{2} + 4 a + 3\right)\cdot 17 + \left(16 a^{2} + 14 a + 16\right)\cdot 17^{2} + \left(16 a^{2} + 7 a + 15\right)\cdot 17^{3} + \left(16 a^{2} + 7 a\right)\cdot 17^{4} + \left(2 a^{2} + 16 a + 8\right)\cdot 17^{5} + \left(6 a^{2} + 12 a + 2\right)\cdot 17^{6} + \left(a^{2} + a + 5\right)\cdot 17^{7} + \left(12 a^{2} + 5 a + 12\right)\cdot 17^{8} + \left(13 a^{2} + 11 a + 13\right)\cdot 17^{9} +O(17^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 13 a^{2} + 12 a + 14 + \left(a^{2} + 6 a + 7\right)\cdot 17 + \left(10 a^{2} + 16 a + 14\right)\cdot 17^{2} + \left(9 a^{2} + 15 a + 5\right)\cdot 17^{3} + \left(16 a^{2} + 14 a + 5\right)\cdot 17^{4} + \left(4 a^{2} + 14 a + 5\right)\cdot 17^{5} + \left(5 a^{2} + 8 a + 5\right)\cdot 17^{6} + \left(a^{2} + 1\right)\cdot 17^{7} + \left(14 a^{2} + 4 a + 8\right)\cdot 17^{8} + \left(12 a^{2} + 3 a + 11\right)\cdot 17^{9} +O(17^{10})\)
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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 |
| $1$ | $1$ | $()$ | $7$ |
| $7$ | $2$ | $(1,6)(2,5)(3,7)(4,8)$ | $-1$ |
| $28$ | $3$ | $(1,4,7)(3,6,8)$ | $\zeta_{3}$ |
| $28$ | $3$ | $(1,7,4)(3,8,6)$ | $-\zeta_{3} - 1$ |
| $28$ | $6$ | $(1,3,4,6,7,8)(2,5)$ | $\zeta_{3} + 1$ |
| $28$ | $6$ | $(1,8,7,6,4,3)(2,5)$ | $-\zeta_{3}$ |
| $24$ | $7$ | $(1,4,6,5,2,8,3)$ | $0$ |
| $24$ | $7$ | $(1,5,3,6,8,4,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.