Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:C_4$ |
| Conductor: | \(3115008\)\(\medspace = 2^{11} \cdot 3^{2} \cdot 13^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.24920064.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_3^2:C_4$ |
| Parity: | even |
| Determinant: | 1.8.2t1.a.a |
| Projective image: | $C_3^2:C_4$ |
| Projective stem field: | Galois closure of 6.2.24920064.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 11x^{4} - 12x^{3} + 25x^{2} - 14x - 1 \)
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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 $\mathbb{Q}_{ 17 }$:
\( x^{2} + 16x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 14 a + 4 + \left(9 a + 1\right)\cdot 17 + \left(12 a + 12\right)\cdot 17^{2} + \left(5 a + 7\right)\cdot 17^{3} + 13 a\cdot 17^{4} + \left(9 a + 6\right)\cdot 17^{5} + \left(8 a + 9\right)\cdot 17^{6} + \left(9 a + 5\right)\cdot 17^{7} + \left(3 a + 8\right)\cdot 17^{8} + 2 a\cdot 17^{9} +O(17^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 13 + 17 + 7\cdot 17^{2} + 8\cdot 17^{3} + 8\cdot 17^{4} + 8\cdot 17^{5} + 16\cdot 17^{6} + 4\cdot 17^{7} + 6\cdot 17^{8} +O(17^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 3 a + 1 + \left(7 a + 14\right)\cdot 17 + \left(4 a + 14\right)\cdot 17^{2} + 11 a\cdot 17^{3} + \left(3 a + 8\right)\cdot 17^{4} + \left(7 a + 2\right)\cdot 17^{5} + \left(8 a + 8\right)\cdot 17^{6} + \left(7 a + 6\right)\cdot 17^{7} + \left(13 a + 2\right)\cdot 17^{8} + \left(14 a + 16\right)\cdot 17^{9} +O(17^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 15 + 14\cdot 17 + 14\cdot 17^{2} + 7\cdot 17^{3} + 7\cdot 17^{4} + 17^{5} + 11\cdot 17^{6} + 8\cdot 17^{8} + 4\cdot 17^{9} +O(17^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 16 a + 2 + \left(5 a + 6\right)\cdot 17 + \left(15 a + 13\right)\cdot 17^{2} + \left(14 a + 4\right)\cdot 17^{3} + 3\cdot 17^{4} + \left(5 a + 14\right)\cdot 17^{5} + 13\cdot 17^{6} + \left(13 a + 1\right)\cdot 17^{7} + \left(10 a + 14\right)\cdot 17^{8} + \left(14 a + 12\right)\cdot 17^{9} +O(17^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( a + 1 + \left(11 a + 13\right)\cdot 17 + \left(a + 5\right)\cdot 17^{2} + \left(2 a + 4\right)\cdot 17^{3} + \left(16 a + 6\right)\cdot 17^{4} + \left(11 a + 1\right)\cdot 17^{5} + \left(16 a + 9\right)\cdot 17^{6} + \left(3 a + 14\right)\cdot 17^{7} + \left(6 a + 11\right)\cdot 17^{8} + \left(2 a + 16\right)\cdot 17^{9} +O(17^{10})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 6 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 6 }$ | Character value | Complex conjugation |
| $1$ | $1$ | $()$ | $4$ | |
| $9$ | $2$ | $(1,2)(4,5)$ | $0$ | ✓ |
| $4$ | $3$ | $(1,2,3)$ | $1$ | |
| $4$ | $3$ | $(1,2,3)(4,5,6)$ | $-2$ | |
| $9$ | $4$ | $(1,5,2,4)(3,6)$ | $0$ | |
| $9$ | $4$ | $(1,4,2,5)(3,6)$ | $0$ |