Basic invariants
| Dimension: | $3$ |
| Group: | $C_7:C_3$ |
| Conductor: | \(134456\)\(\medspace = 2^{3} \cdot 7^{5} \) |
| Artin stem field: | Galois closure of 7.7.18078415936.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_7:C_3$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $C_7:C_3$ |
| Projective stem field: | Galois closure of 7.7.18078415936.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 14x^{5} + 56x^{3} - 56x - 22 \)
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The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$:
\( x^{3} + 2x + 9 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 9\cdot 11 + 9\cdot 11^{2} + 8\cdot 11^{3} + 11^{4} + 3\cdot 11^{5} + 6\cdot 11^{6} + 5\cdot 11^{7} + 2\cdot 11^{8} + 3\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 7 a^{2} + 7 a + 4 + \left(9 a^{2} + 2 a + 5\right)\cdot 11 + \left(4 a^{2} + 9 a + 5\right)\cdot 11^{2} + \left(6 a^{2} + 8 a + 6\right)\cdot 11^{3} + \left(2 a^{2} + 7 a + 4\right)\cdot 11^{4} + \left(7 a^{2} + 5 a\right)\cdot 11^{5} + \left(10 a^{2} + 10 a + 10\right)\cdot 11^{6} + \left(2 a^{2} + 8 a + 4\right)\cdot 11^{7} + \left(6 a^{2} + 4 a + 4\right)\cdot 11^{8} + \left(2 a^{2} + 10 a + 1\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 4 a^{2} + 8 a + 8 a^{2} 11 + \left(a^{2} + 5 a + 5\right)\cdot 11^{2} + \left(2 a^{2} + 9 a + 4\right)\cdot 11^{3} + \left(4 a^{2} + 8 a + 10\right)\cdot 11^{4} + \left(10 a^{2} + 8 a\right)\cdot 11^{5} + \left(6 a^{2} + 10 a + 5\right)\cdot 11^{6} + \left(9 a^{2} + 8 a + 6\right)\cdot 11^{7} + \left(7 a^{2} + 5 a + 6\right)\cdot 11^{8} + \left(4 a^{2} + 6 a\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 4 a + 9 + \left(6 a^{2} + 6 a + 8\right)\cdot 11 + \left(7 a^{2} + 3 a + 7\right)\cdot 11^{2} + \left(5 a^{2} + 8 a + 6\right)\cdot 11^{3} + \left(4 a^{2} + 10 a\right)\cdot 11^{4} + \left(5 a^{2} + 4 a + 8\right)\cdot 11^{5} + \left(5 a^{2} + 5 a + 5\right)\cdot 11^{6} + \left(10 a^{2} + 2 a + 7\right)\cdot 11^{7} + \left(9 a^{2} + 7 a + 1\right)\cdot 11^{8} + \left(5 a^{2} + 7 a + 5\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 4 a^{2} + 4 a + 7 + \left(6 a^{2} + 3 a + 5\right)\cdot 11 + \left(3 a^{2} + 7 a + 2\right)\cdot 11^{2} + \left(3 a^{2} + 5 a + 7\right)\cdot 11^{3} + \left(10 a^{2} + 8 a + 4\right)\cdot 11^{4} + \left(4 a^{2} + 5 a + 7\right)\cdot 11^{5} + \left(5 a^{2} + 10 a + 5\right)\cdot 11^{6} + \left(2 a^{2} + 4 a\right)\cdot 11^{7} + 3 a^{2} 11^{8} + \left(7 a^{2} + 5 a + 7\right)\cdot 11^{9} +O(11^{10})\)
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| $r_{ 6 }$ | $=$ |
\( 7 a^{2} + 3 a + \left(9 a^{2} + a + 10\right)\cdot 11 + \left(10 a^{2} + 4\right)\cdot 11^{2} + \left(a^{2} + 8 a + 5\right)\cdot 11^{3} + \left(7 a^{2} + 2 a\right)\cdot 11^{4} + 9\cdot 11^{5} + \left(6 a + 5\right)\cdot 11^{6} + \left(9 a^{2} + 3 a + 5\right)\cdot 11^{7} + \left(8 a^{2} + 3 a + 7\right)\cdot 11^{8} + \left(8 a^{2} + 9 a + 1\right)\cdot 11^{9} +O(11^{10})\)
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| $r_{ 7 }$ | $=$ |
\( 7 a + 2 + \left(4 a^{2} + 7 a + 5\right)\cdot 11 + \left(4 a^{2} + 7 a + 8\right)\cdot 11^{2} + \left(2 a^{2} + 3 a + 4\right)\cdot 11^{3} + \left(4 a^{2} + 5 a + 10\right)\cdot 11^{4} + \left(4 a^{2} + 7 a + 3\right)\cdot 11^{5} + \left(4 a^{2} + 5\right)\cdot 11^{6} + \left(9 a^{2} + 4 a + 2\right)\cdot 11^{7} + \left(7 a^{2} + 10\right)\cdot 11^{8} + \left(3 a^{2} + 5 a + 2\right)\cdot 11^{9} +O(11^{10})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 7 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 7 }$ | Character value |
| $1$ | $1$ | $()$ | $3$ |
| $7$ | $3$ | $(1,6,2)(4,7,5)$ | $0$ |
| $7$ | $3$ | $(1,2,6)(4,5,7)$ | $0$ |
| $3$ | $7$ | $(1,5,6,7,4,3,2)$ | $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ |
| $3$ | $7$ | $(1,7,2,6,3,5,4)$ | $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.