Basic invariants
| Dimension: | $2$ |
| Group: | $C_4\wr C_2$ |
| Conductor: | \(416\)\(\medspace = 2^{5} \cdot 13 \) |
| Artin number field: | Galois closure of 8.0.143982592.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4\wr C_2$ |
| Parity: | odd |
| Projective image: | $D_4$ |
| Projective field: | Galois closure of 4.2.35152.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$:
\( x^{2} + 21x + 5 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 15 a + 5 + \left(9 a + 12\right)\cdot 23 + \left(5 a + 6\right)\cdot 23^{2} + \left(21 a + 14\right)\cdot 23^{3} + \left(9 a + 18\right)\cdot 23^{4} + \left(19 a + 17\right)\cdot 23^{5} + \left(4 a + 12\right)\cdot 23^{6} + \left(3 a + 17\right)\cdot 23^{7} + \left(8 a + 6\right)\cdot 23^{8} + 20\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 10 a + 20 + \left(17 a + 4\right)\cdot 23 + \left(10 a + 3\right)\cdot 23^{2} + \left(15 a + 13\right)\cdot 23^{3} + \left(6 a + 5\right)\cdot 23^{4} + \left(14 a + 10\right)\cdot 23^{5} + \left(2 a + 19\right)\cdot 23^{6} + \left(7 a + 3\right)\cdot 23^{7} + \left(18 a + 11\right)\cdot 23^{8} + \left(14 a + 10\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 10 a + 6 + \left(17 a + 16\right)\cdot 23 + \left(10 a + 15\right)\cdot 23^{2} + \left(15 a + 12\right)\cdot 23^{3} + \left(6 a + 19\right)\cdot 23^{4} + \left(14 a + 13\right)\cdot 23^{5} + \left(2 a + 12\right)\cdot 23^{6} + \left(7 a + 7\right)\cdot 23^{7} + \left(18 a + 5\right)\cdot 23^{8} + \left(14 a + 1\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 15 a + 11 + \left(9 a + 6\right)\cdot 23 + \left(5 a + 15\right)\cdot 23^{2} + \left(21 a + 17\right)\cdot 23^{3} + \left(9 a + 5\right)\cdot 23^{4} + \left(19 a + 22\right)\cdot 23^{5} + \left(4 a + 19\right)\cdot 23^{6} + \left(3 a + 3\right)\cdot 23^{7} + \left(8 a + 3\right)\cdot 23^{8} + 10\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 8 a + 18 + \left(13 a + 10\right)\cdot 23 + \left(17 a + 16\right)\cdot 23^{2} + \left(a + 8\right)\cdot 23^{3} + \left(13 a + 4\right)\cdot 23^{4} + \left(3 a + 5\right)\cdot 23^{5} + \left(18 a + 10\right)\cdot 23^{6} + \left(19 a + 5\right)\cdot 23^{7} + \left(14 a + 16\right)\cdot 23^{8} + \left(22 a + 2\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 13 a + 3 + \left(5 a + 18\right)\cdot 23 + \left(12 a + 19\right)\cdot 23^{2} + \left(7 a + 9\right)\cdot 23^{3} + \left(16 a + 17\right)\cdot 23^{4} + \left(8 a + 12\right)\cdot 23^{5} + \left(20 a + 3\right)\cdot 23^{6} + \left(15 a + 19\right)\cdot 23^{7} + \left(4 a + 11\right)\cdot 23^{8} + \left(8 a + 12\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 13 a + 17 + \left(5 a + 6\right)\cdot 23 + \left(12 a + 7\right)\cdot 23^{2} + \left(7 a + 10\right)\cdot 23^{3} + \left(16 a + 3\right)\cdot 23^{4} + \left(8 a + 9\right)\cdot 23^{5} + \left(20 a + 10\right)\cdot 23^{6} + \left(15 a + 15\right)\cdot 23^{7} + \left(4 a + 17\right)\cdot 23^{8} + \left(8 a + 21\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 8 a + 12 + \left(13 a + 16\right)\cdot 23 + \left(17 a + 7\right)\cdot 23^{2} + \left(a + 5\right)\cdot 23^{3} + \left(13 a + 17\right)\cdot 23^{4} + 3 a\cdot 23^{5} + \left(18 a + 3\right)\cdot 23^{6} + \left(19 a + 19\right)\cdot 23^{7} + \left(14 a + 19\right)\cdot 23^{8} + \left(22 a + 12\right)\cdot 23^{9} +O(23^{10})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 8 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 8 }$ | Character values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $-2$ | $-2$ |
| $2$ | $2$ | $(1,5)(3,7)$ | $0$ | $0$ |
| $4$ | $2$ | $(1,2)(3,8)(4,7)(5,6)$ | $0$ | $0$ |
| $1$ | $4$ | $(1,3,5,7)(2,8,6,4)$ | $-2 \zeta_{4}$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,7,5,3)(2,4,6,8)$ | $2 \zeta_{4}$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,7,5,3)(2,8,6,4)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,3,5,7)$ | $-\zeta_{4} + 1$ | $\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,7,5,3)$ | $\zeta_{4} + 1$ | $-\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,7,5,3)(2,6)(4,8)$ | $\zeta_{4} - 1$ | $-\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,3,5,7)(2,6)(4,8)$ | $-\zeta_{4} - 1$ | $\zeta_{4} - 1$ |
| $4$ | $4$ | $(1,2,5,6)(3,8,7,4)$ | $0$ | $0$ |
| $4$ | $8$ | $(1,2,3,8,5,6,7,4)$ | $0$ | $0$ |
| $4$ | $8$ | $(1,8,7,2,5,4,3,6)$ | $0$ | $0$ |