Basic invariants
| Dimension: | $1$ |
| Group: | $C_7$ |
| Conductor: | \(29\) |
| Artin number field: | Galois closure of 7.7.594823321.1 |
| Galois orbit size: | $6$ |
| Smallest permutation container: | $C_7$ |
| Parity: | even |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$:
\( x^{7} + 4x + 9 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( a^{6} + 3 a^{5} + 3 a^{4} + 5 a^{3} + 10 a^{2} + 8 a + 2 + \left(9 a^{6} + 7 a^{5} + 6 a^{4} + 6 a^{3} + 10 a^{2} + 3 a + 9\right)\cdot 11 + \left(5 a^{6} + 3 a^{5} + 8 a^{4} + a^{3} + 10 a^{2} + 4 a + 5\right)\cdot 11^{2} + \left(8 a^{6} + 9 a^{4} + 8 a^{3} + 5 a^{2} + 5 a + 2\right)\cdot 11^{3} + \left(8 a^{6} + 4 a^{5} + 6 a^{4} + 5 a^{3} + 4 a^{2} + 7 a\right)\cdot 11^{4} + \left(8 a^{6} + 7 a^{5} + 3 a^{4} + 6 a^{3} + 3 a^{2} + a + 5\right)\cdot 11^{5} +O(11^{6})\)
| $r_{ 2 }$ |
$=$ |
\( 5 a^{6} + 2 a^{5} + 8 a^{4} + 2 a^{3} + 6 a^{2} + 9 a + \left(6 a^{6} + 7 a^{5} + 6 a^{4} + 4 a^{3} + 10 a^{2} + 2 a + 8\right)\cdot 11 + \left(4 a^{6} + 3 a^{4} + 9 a^{3} + 7 a^{2} + 2 a + 4\right)\cdot 11^{2} + \left(9 a^{6} + 5 a^{5} + a^{4} + 5 a^{3} + 10 a^{2} + 2\right)\cdot 11^{3} + \left(8 a^{6} + a^{5} + 6 a^{4} + 2 a^{3} + 8 a^{2} + 8 a + 5\right)\cdot 11^{4} + \left(10 a^{6} + 6 a^{5} + 5 a^{4} + 6 a^{2} + 3 a + 2\right)\cdot 11^{5} +O(11^{6})\)
| $r_{ 3 }$ |
$=$ |
\( 6 a^{6} + 10 a^{5} + 9 a^{4} + 10 a^{3} + 4 a^{2} + 7 a + 5 + \left(9 a^{6} + 4 a^{5} + 5 a^{4} + 7 a^{3} + 7 a^{2} + 8 a + 4\right)\cdot 11 + \left(10 a^{6} + a^{5} + 10 a^{4} + 10 a^{3} + 3 a^{2} + 5 a + 7\right)\cdot 11^{2} + \left(10 a^{5} + a^{4} + 6 a^{2} + 3 a + 9\right)\cdot 11^{3} + \left(3 a^{5} + 10 a^{4} + 9 a^{2} + 5 a + 1\right)\cdot 11^{4} + \left(3 a^{6} + 2 a^{5} + 7 a^{4} + 6 a^{3} + 7 a^{2} + 8 a + 7\right)\cdot 11^{5} +O(11^{6})\)
| $r_{ 4 }$ |
$=$ |
\( 7 a^{6} + 10 a^{4} + 6 a^{3} + 7 a^{2} + 7 a + 10 + \left(9 a^{5} + 5 a^{4} + 2 a^{3} + 10 a^{2} + 9 a + 6\right)\cdot 11 + \left(10 a^{5} + 8 a^{4} + 8 a^{3} + 2 a^{2} + 5 a + 9\right)\cdot 11^{2} + \left(3 a^{6} + 6 a^{4} + 4 a^{3} + 10 a^{2}\right)\cdot 11^{3} + \left(2 a^{6} + 9 a^{5} + 10 a^{4} + 4 a^{2} + a + 3\right)\cdot 11^{4} + \left(10 a^{6} + 8 a^{5} + 8 a^{4} + 2 a^{3} + a^{2} + 6 a + 8\right)\cdot 11^{5} +O(11^{6})\)
| $r_{ 5 }$ |
$=$ |
\( 7 a^{6} + 8 a^{5} + 7 a^{4} + 7 a^{3} + 2 a^{2} + 10 + \left(7 a^{6} + 3 a^{5} + 4 a^{3} + 4 a^{2} + 7 a + 8\right)\cdot 11 + \left(10 a^{6} + 5 a^{5} + 6 a^{4} + 9 a^{3} + 6 a^{2} + 5 a + 6\right)\cdot 11^{2} + \left(3 a^{6} + 7 a^{5} + 6 a^{4} + 3 a^{3} + a + 2\right)\cdot 11^{3} + \left(3 a^{6} + 6 a^{5} + 3 a^{3} + 3 a^{2} + 4 a + 5\right)\cdot 11^{4} + \left(3 a^{6} + 8 a^{5} + 9 a^{4} + 3 a^{3} + 3 a + 3\right)\cdot 11^{5} +O(11^{6})\)
| $r_{ 6 }$ |
$=$ |
\( 9 a^{6} + 3 a^{5} + 4 a^{4} + 2 a^{2} + 6 a + 9 + \left(8 a^{6} + 4 a^{5} + 2 a^{4} + a^{3} + 10 a^{2} + 2 a + 9\right)\cdot 11 + \left(9 a^{6} + 5 a^{4} + 6 a^{3} + 8 a^{2} + 10 a + 6\right)\cdot 11^{2} + \left(2 a^{6} + 10 a^{5} + 9 a^{4} + 4 a^{2} + 8 a + 3\right)\cdot 11^{3} + \left(7 a^{6} + 10 a^{4} + a^{3} + 8 a^{2} + 4 a + 1\right)\cdot 11^{4} + \left(7 a^{6} + 6 a^{5} + 5 a^{4} + 7 a^{3} + 9 a^{2} + 8 a + 1\right)\cdot 11^{5} +O(11^{6})\)
| $r_{ 7 }$ |
$=$ |
\( 9 a^{6} + 7 a^{5} + 3 a^{4} + 3 a^{3} + 2 a^{2} + 7 a + 9 + \left(a^{6} + 7 a^{5} + 5 a^{4} + 6 a^{3} + a^{2} + 9 a + 7\right)\cdot 11 + \left(2 a^{6} + 10 a^{5} + a^{4} + 9 a^{3} + 3 a^{2} + 9 a + 2\right)\cdot 11^{2} + \left(4 a^{6} + 9 a^{5} + 8 a^{4} + 8 a^{3} + 5 a^{2} + a\right)\cdot 11^{3} + \left(2 a^{6} + 6 a^{5} + 9 a^{4} + 8 a^{3} + 4 a^{2} + 2 a + 5\right)\cdot 11^{4} + \left(4 a^{5} + 2 a^{4} + 7 a^{3} + 3 a^{2} + a + 5\right)\cdot 11^{5} +O(11^{6})\)
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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 values | |||||
| $c1$ | $c2$ | $c3$ | $c4$ | $c5$ | $c6$ | |||
| $1$ | $1$ | $()$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ |
| $1$ | $7$ | $(1,5,3,6,7,2,4)$ | $\zeta_{7}$ | $\zeta_{7}^{2}$ | $\zeta_{7}^{3}$ | $\zeta_{7}^{4}$ | $\zeta_{7}^{5}$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
| $1$ | $7$ | $(1,3,7,4,5,6,2)$ | $\zeta_{7}^{2}$ | $\zeta_{7}^{4}$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$ | $\zeta_{7}$ | $\zeta_{7}^{3}$ | $\zeta_{7}^{5}$ |
| $1$ | $7$ | $(1,6,4,3,2,5,7)$ | $\zeta_{7}^{3}$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$ | $\zeta_{7}^{2}$ | $\zeta_{7}^{5}$ | $\zeta_{7}$ | $\zeta_{7}^{4}$ |
| $1$ | $7$ | $(1,7,5,2,3,4,6)$ | $\zeta_{7}^{4}$ | $\zeta_{7}$ | $\zeta_{7}^{5}$ | $\zeta_{7}^{2}$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$ | $\zeta_{7}^{3}$ |
| $1$ | $7$ | $(1,2,6,5,4,7,3)$ | $\zeta_{7}^{5}$ | $\zeta_{7}^{3}$ | $\zeta_{7}$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$ | $\zeta_{7}^{4}$ | $\zeta_{7}^{2}$ |
| $1$ | $7$ | $(1,4,2,7,6,3,5)$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$ | $\zeta_{7}^{5}$ | $\zeta_{7}^{4}$ | $\zeta_{7}^{3}$ | $\zeta_{7}^{2}$ | $\zeta_{7}$ |