Basic invariants
| Dimension: | $3$ |
| Group: | $C_7:C_3$ |
| Conductor: | \(7634169\)\(\medspace = 3^{4} \cdot 307^{2} \) |
| Artin stem field: | Galois closure of 7.7.58280536320561.1 |
| 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 7.7.58280536320561.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 3x^{6} - 36x^{5} + 143x^{4} + 204x^{3} - 1476x^{2} + 1800x - 432 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 9 a^{2} + 8 a + 6 + \left(6 a^{2} + a + 9\right)\cdot 11 + \left(8 a^{2} + 7 a + 10\right)\cdot 11^{2} + \left(5 a^{2} + a + 10\right)\cdot 11^{3} + \left(8 a^{2} + 7\right)\cdot 11^{4} + \left(a^{2} + 8 a + 6\right)\cdot 11^{5} + \left(3 a + 9\right)\cdot 11^{6} + \left(5 a^{2} + 2 a + 4\right)\cdot 11^{7} + \left(10 a^{2} + 5 a + 9\right)\cdot 11^{8} + \left(2 a^{2} + a + 3\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 5 a^{2} + 5 a + 7 + \left(2 a^{2} + 6 a + 10\right)\cdot 11 + \left(8 a^{2} + 8 a + 1\right)\cdot 11^{2} + \left(8 a^{2} + a + 9\right)\cdot 11^{3} + \left(7 a^{2} + 10 a + 5\right)\cdot 11^{4} + \left(a^{2} + 4 a\right)\cdot 11^{5} + \left(10 a^{2} + 8 a + 3\right)\cdot 11^{6} + \left(6 a^{2} + 7 a + 1\right)\cdot 11^{7} + \left(8 a^{2} + 7 a + 6\right)\cdot 11^{8} + \left(7 a + 1\right)\cdot 11^{9} +O(11^{10})\)
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| $r_{ 3 }$ | $=$ |
\( a^{2} + 10 + \left(a^{2} + 10 a + 1\right)\cdot 11 + \left(7 a^{2} + 5\right)\cdot 11^{2} + \left(4 a^{2} + 5 a + 9\right)\cdot 11^{3} + \left(2 a^{2} + 2 a + 10\right)\cdot 11^{4} + \left(6 a^{2} + 8 a + 8\right)\cdot 11^{5} + \left(4 a^{2} + 6 a\right)\cdot 11^{6} + \left(9 a^{2} + 7\right)\cdot 11^{7} + \left(9 a^{2} + a + 8\right)\cdot 11^{8} + \left(4 a^{2} + 2\right)\cdot 11^{9} +O(11^{10})\)
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| $r_{ 4 }$ | $=$ |
\( 5 a + 4 + \left(8 a^{2} + 3 a + 3\right)\cdot 11 + \left(9 a^{2} + 6 a\right)\cdot 11^{2} + \left(8 a^{2} + 2 a + 2\right)\cdot 11^{3} + \left(3 a^{2} + 5 a + 4\right)\cdot 11^{4} + \left(6 a^{2} + 6 a + 10\right)\cdot 11^{5} + \left(7 a^{2} + 9 a + 6\right)\cdot 11^{6} + \left(4 a^{2} + 7 a + 5\right)\cdot 11^{7} + \left(5 a^{2} + 10 a + 5\right)\cdot 11^{8} + \left(2 a^{2} + 10 a + 7\right)\cdot 11^{9} +O(11^{10})\)
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| $r_{ 5 }$ | $=$ |
\( a^{2} + 3 a + 10 + \left(3 a^{2} + 10 a\right)\cdot 11 + \left(6 a^{2} + 2 a + 4\right)\cdot 11^{2} + 4 a\cdot 11^{3} + \left(8 a + 4\right)\cdot 11^{4} + \left(3 a^{2} + 5 a + 8\right)\cdot 11^{5} + \left(6 a^{2} + 6\right)\cdot 11^{6} + \left(7 a^{2} + 8 a + 4\right)\cdot 11^{7} + \left(a^{2} + 4 a + 1\right)\cdot 11^{8} + \left(3 a^{2} + 9 a + 4\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 6 a^{2} + a + 1 + \left(a + 8\right)\cdot 11 + \left(4 a^{2} + 7 a + 3\right)\cdot 11^{2} + \left(4 a^{2} + 6 a + 3\right)\cdot 11^{3} + \left(10 a^{2} + 6 a + 9\right)\cdot 11^{4} + \left(2 a^{2} + 10 a + 5\right)\cdot 11^{5} + \left(4 a^{2} + 3 a + 2\right)\cdot 11^{6} + \left(10 a^{2} + 6 a + 2\right)\cdot 11^{7} + \left(7 a^{2} + 3 a + 5\right)\cdot 11^{8} + \left(7 a^{2} + 3 a + 3\right)\cdot 11^{9} +O(11^{10})\)
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| $r_{ 7 }$ | $=$ |
\( 9 + 9\cdot 11 + 6\cdot 11^{2} + 8\cdot 11^{3} + 11^{4} + 3\cdot 11^{5} + 3\cdot 11^{6} + 7\cdot 11^{7} + 7\cdot 11^{8} + 9\cdot 11^{9} +O(11^{10})\)
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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 | Complex conjugation |
| $1$ | $1$ | $()$ | $3$ | ✓ |
| $7$ | $3$ | $(1,3,5)(2,4,6)$ | $0$ | |
| $7$ | $3$ | $(1,5,3)(2,6,4)$ | $0$ | |
| $3$ | $7$ | $(1,4,5,3,7,6,2)$ | $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ | |
| $3$ | $7$ | $(1,3,2,5,6,4,7)$ | $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ |