Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:C_4$ |
| Conductor: | \(962948\)\(\medspace = 2^{2} \cdot 7^{2} \cdot 17^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.65480464.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_3^2:C_4$ |
| Parity: | even |
| Determinant: | 1.17.2t1.a.a |
| Projective image: | $C_3^2:C_4$ |
| Projective stem field: | Galois closure of 6.2.65480464.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + x^{4} - 6x^{3} - 11x^{2} + 9 \)
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The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$:
\( x^{2} + 12x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 10 a + 4 + \left(9 a + 8\right)\cdot 13 + \left(3 a + 5\right)\cdot 13^{2} + \left(9 a + 9\right)\cdot 13^{3} + \left(7 a + 6\right)\cdot 13^{4} + \left(4 a + 10\right)\cdot 13^{5} + \left(8 a + 11\right)\cdot 13^{6} + \left(a + 6\right)\cdot 13^{7} + \left(12 a + 3\right)\cdot 13^{8} + \left(8 a + 10\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 10 a + 11 + \left(5 a + 8\right)\cdot 13 + \left(6 a + 3\right)\cdot 13^{2} + \left(2 a + 2\right)\cdot 13^{3} + \left(4 a + 4\right)\cdot 13^{4} + \left(2 a + 1\right)\cdot 13^{5} + \left(a + 2\right)\cdot 13^{6} + \left(8 a + 6\right)\cdot 13^{7} + \left(4 a + 10\right)\cdot 13^{8} + \left(11 a + 5\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 8 + 12\cdot 13 + 4\cdot 13^{2} + 12\cdot 13^{3} + 2\cdot 13^{4} + 12\cdot 13^{5} + 9\cdot 13^{6} + 6\cdot 13^{7} + 8\cdot 13^{8} + 7\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 3 a + 1 + \left(3 a + 8\right)\cdot 13 + \left(9 a + 12\right)\cdot 13^{2} + \left(3 a + 1\right)\cdot 13^{3} + \left(5 a + 5\right)\cdot 13^{4} + \left(8 a + 7\right)\cdot 13^{5} + \left(4 a + 2\right)\cdot 13^{6} + 11 a\cdot 13^{7} + 13^{8} + \left(4 a + 7\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 9 + 9\cdot 13 + 7\cdot 13^{2} + 13^{3} + 13^{4} + 8\cdot 13^{5} + 11\cdot 13^{6} + 5\cdot 13^{7} + 8\cdot 13^{8} + 8\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 3 a + 8 + \left(7 a + 4\right)\cdot 13 + \left(6 a + 4\right)\cdot 13^{2} + \left(10 a + 11\right)\cdot 13^{3} + \left(8 a + 5\right)\cdot 13^{4} + \left(10 a + 12\right)\cdot 13^{5} + 11 a\cdot 13^{6} + 4 a\cdot 13^{7} + \left(8 a + 7\right)\cdot 13^{8} + \left(a + 12\right)\cdot 13^{9} +O(13^{10})\)
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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 |
| $1$ | $1$ | $()$ | $4$ |
| $9$ | $2$ | $(1,4)(2,3)$ | $0$ |
| $4$ | $3$ | $(2,3,6)$ | $-2$ |
| $4$ | $3$ | $(1,4,5)(2,3,6)$ | $1$ |
| $9$ | $4$ | $(1,3,4,2)(5,6)$ | $0$ |
| $9$ | $4$ | $(1,2,4,3)(5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.