Basic invariants
| Dimension: | $3$ |
| Group: | $C_7:C_3$ |
| Conductor: | \(769129\)\(\medspace = 877^{2} \) |
| Artin stem field: | Galois closure of 7.7.591559418641.1 |
| 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 7.7.591559418641.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 2x^{6} - 20x^{5} + 34x^{4} + 107x^{3} - 166x^{2} - 119x + 138 \)
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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 $\mathbb{Q}_{ 13 }$:
\( x^{3} + 2x + 11 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( a^{2} + 9 a + 8 + \left(3 a^{2} + 12\right)\cdot 13 + \left(11 a^{2} + 2 a + 4\right)\cdot 13^{2} + \left(2 a^{2} + a + 4\right)\cdot 13^{3} + \left(12 a^{2} + 5 a + 12\right)\cdot 13^{4} + \left(10 a^{2} + 6 a\right)\cdot 13^{5} + \left(7 a^{2} + 2 a + 1\right)\cdot 13^{6} + \left(3 a^{2} + 12 a + 7\right)\cdot 13^{7} + \left(6 a^{2} + 7 a + 8\right)\cdot 13^{8} + \left(9 a^{2} + 9 a + 7\right)\cdot 13^{9} +O(13^{10})\)
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| $r_{ 2 }$ | $=$ |
\( 7 a^{2} + 5 a + 10 + \left(11 a^{2} + a + 1\right)\cdot 13 + \left(12 a^{2} + 12 a + 8\right)\cdot 13^{2} + \left(a^{2} + 9 a + 12\right)\cdot 13^{3} + \left(a^{2} + 12 a + 1\right)\cdot 13^{4} + \left(2 a + 2\right)\cdot 13^{5} + \left(a^{2} + 4 a + 5\right)\cdot 13^{6} + \left(a^{2} + 9 a + 2\right)\cdot 13^{7} + \left(12 a^{2} + 2 a + 5\right)\cdot 13^{8} + \left(12 a^{2} + 11 a + 3\right)\cdot 13^{9} +O(13^{10})\)
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| $r_{ 3 }$ | $=$ |
\( 6 + 2\cdot 13 + 5\cdot 13^{2} + 7\cdot 13^{3} + 10\cdot 13^{4} + 8\cdot 13^{5} + 3\cdot 13^{6} + 3\cdot 13^{7} + 6\cdot 13^{8} + 4\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( a^{2} + 7 a + 8 + \left(7 a^{2} + 3 a\right)\cdot 13 + \left(5 a^{2} + 6\right)\cdot 13^{2} + \left(9 a^{2} + 4\right)\cdot 13^{3} + \left(12 a + 1\right)\cdot 13^{4} + \left(3 a + 8\right)\cdot 13^{5} + \left(6 a^{2} + 5 a + 11\right)\cdot 13^{6} + \left(9 a^{2} + 11 a + 1\right)\cdot 13^{7} + \left(11 a^{2} + 5 a + 7\right)\cdot 13^{8} + \left(6 a^{2} + 3 a + 8\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 11 a^{2} + 6 a + 11 + \left(7 a^{2} + 10 a + 9\right)\cdot 13 + 10 a^{2} 13^{2} + \left(3 a^{2} + 9 a + 2\right)\cdot 13^{3} + \left(9 a^{2} + 7 a + 4\right)\cdot 13^{4} + \left(5 a^{2} + 4 a + 5\right)\cdot 13^{5} + \left(3 a^{2} + 5 a + 8\right)\cdot 13^{6} + \left(11 a^{2} + 12 a + 11\right)\cdot 13^{7} + \left(2 a^{2} + 7 a + 5\right)\cdot 13^{8} + \left(11 a^{2} + 10 a + 5\right)\cdot 13^{9} +O(13^{10})\)
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| $r_{ 6 }$ | $=$ |
\( 11 a^{2} + 10 a + 4 + \left(2 a^{2} + 8 a + 12\right)\cdot 13 + \left(9 a^{2} + 10 a + 10\right)\cdot 13^{2} + \left(11 a + 5\right)\cdot 13^{3} + 8 a\cdot 13^{4} + \left(2 a^{2} + 2 a + 2\right)\cdot 13^{5} + \left(12 a^{2} + 5 a + 11\right)\cdot 13^{6} + \left(12 a^{2} + 2 a + 10\right)\cdot 13^{7} + \left(7 a^{2} + 12 a + 10\right)\cdot 13^{8} + \left(9 a^{2} + 12 a + 7\right)\cdot 13^{9} +O(13^{10})\)
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| $r_{ 7 }$ | $=$ |
\( 8 a^{2} + 2 a + 7 + \left(6 a^{2} + a + 12\right)\cdot 13 + \left(2 a^{2} + 2\right)\cdot 13^{2} + \left(7 a^{2} + 7 a + 2\right)\cdot 13^{3} + \left(2 a^{2} + 5 a + 8\right)\cdot 13^{4} + \left(7 a^{2} + 5 a + 11\right)\cdot 13^{5} + \left(8 a^{2} + 3 a + 10\right)\cdot 13^{6} + \left(4 a + 1\right)\cdot 13^{7} + \left(11 a^{2} + 2 a + 8\right)\cdot 13^{8} + \left(a^{2} + 4 a + 1\right)\cdot 13^{9} +O(13^{10})\)
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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 |
| $1$ | $1$ | $()$ | $3$ |
| $7$ | $3$ | $(1,4,7)(3,6,5)$ | $0$ |
| $7$ | $3$ | $(1,7,4)(3,5,6)$ | $0$ |
| $3$ | $7$ | $(1,4,3,7,5,6,2)$ | $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ |
| $3$ | $7$ | $(1,7,2,3,6,4,5)$ | $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.