Basic invariants
Dimension: | $1$ |
Group: | $C_{12}$ |
Conductor: | \(45\)\(\medspace = 3^{2} \cdot 5 \) |
Artin number field: | Galois closure of 12.0.84075626953125.1 |
Galois orbit size: | $4$ |
Smallest permutation container: | $C_{12}$ |
Parity: | odd |
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 $\Q_{ 17 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$:
\( x^{4} + 7x^{2} + 10x + 3 \)
Roots:
$r_{ 1 }$ | $=$ |
\( a^{3} + 16 a^{2} + 5 + \left(13 a + 10\right)\cdot 17 + \left(9 a^{3} + 12 a^{2} + 9 a + 2\right)\cdot 17^{2} + \left(4 a^{3} + 6 a^{2} + 5 a + 9\right)\cdot 17^{3} + \left(4 a^{2} + 4 a + 8\right)\cdot 17^{4} +O(17^{5})\)
|
$r_{ 2 }$ | $=$ |
\( 12 a^{3} + a^{2} + 5 a + 11 + \left(5 a^{3} + 10 a^{2} + 5 a + 4\right)\cdot 17 + \left(3 a^{3} + 12 a^{2} + 7 a + 15\right)\cdot 17^{2} + \left(7 a^{3} + 14 a^{2} + 9 a + 9\right)\cdot 17^{3} + \left(11 a^{3} + 8 a^{2} + 13 a + 9\right)\cdot 17^{4} +O(17^{5})\)
|
$r_{ 3 }$ | $=$ |
\( 8 a^{3} + 11 a^{2} + 11 a + 10 + \left(14 a^{3} + 6 a^{2} + 11\right)\cdot 17 + \left(13 a^{3} + 6 a^{2} + 5 a + 6\right)\cdot 17^{2} + \left(14 a^{3} + 11 a^{2} + 7 a + 14\right)\cdot 17^{3} + \left(13 a^{3} + 12 a^{2} + a\right)\cdot 17^{4} +O(17^{5})\)
|
$r_{ 4 }$ | $=$ |
\( 4 a^{3} + 14 a^{2} + 14 a + 5 + \left(13 a^{3} + 15 a^{2} + 12 a + 13\right)\cdot 17 + \left(11 a^{3} + 16 a^{2} + 11 a + 8\right)\cdot 17^{2} + \left(5 a^{3} + 15 a^{2} + 4 a + 11\right)\cdot 17^{3} + \left(a^{2} + 13 a + 12\right)\cdot 17^{4} +O(17^{5})\)
|
$r_{ 5 }$ | $=$ |
\( 5 a^{3} + 12 a^{2} + 8 + \left(4 a^{2} + 14 a\right)\cdot 17 + \left(2 a^{3} + a^{2} + 14 a + 2\right)\cdot 17^{2} + \left(16 a^{3} + 14 a^{2} + 12 a + 8\right)\cdot 17^{3} + \left(13 a^{3} + 8 a + 2\right)\cdot 17^{4} +O(17^{5})\)
|
$r_{ 6 }$ | $=$ |
\( 13 a^{3} + 12 a^{2} + 6 a + 13 + \left(8 a^{3} + 14 a^{2} + a + 4\right)\cdot 17 + \left(14 a^{3} + 8 a + 13\right)\cdot 17^{2} + \left(7 a^{3} + 13 a^{2} + 4 a + 13\right)\cdot 17^{3} + \left(13 a^{3} + a^{2} + 15 a + 4\right)\cdot 17^{4} +O(17^{5})\)
|
$r_{ 7 }$ | $=$ |
\( 11 a^{3} + 6 a^{2} + 4 + \left(16 a^{3} + 11 a^{2} + 7 a + 6\right)\cdot 17 + \left(5 a^{3} + 3 a^{2} + 9 a + 12\right)\cdot 17^{2} + \left(13 a^{3} + 13 a^{2} + 15 a + 16\right)\cdot 17^{3} + \left(2 a^{3} + 11 a^{2} + 3 a + 5\right)\cdot 17^{4} +O(17^{5})\)
|
$r_{ 8 }$ | $=$ |
\( 15 a^{3} + 14 a^{2} + 2 a + 1 + \left(a^{3} + 6 a^{2} + 16 a + 2\right)\cdot 17 + \left(2 a^{3} + 10 a^{2} + 15 a + 13\right)\cdot 17^{2} + \left(8 a^{3} + 7 a^{2} + a + 13\right)\cdot 17^{3} + \left(8 a^{2} + 12 a + 15\right)\cdot 17^{4} +O(17^{5})\)
|
$r_{ 9 }$ | $=$ |
\( 14 a^{3} + 9 a^{2} + 13 a + 14 + \left(9 a^{3} + 5 a^{2} + 6 a + 6\right)\cdot 17 + \left(6 a^{3} + 15 a^{2} + 3 a\right)\cdot 17^{2} + \left(a^{3} + 13 a^{2} + 8 a + 16\right)\cdot 17^{3} + \left(a^{3} + 3 a^{2} + 15 a + 9\right)\cdot 17^{4} +O(17^{5})\)
|
$r_{ 10 }$ | $=$ |
\( 5 a^{3} + 9 a^{2} + 9 a + 2 + \left(6 a^{3} + 11 a^{2} + 3 a + 9\right)\cdot 17 + \left(8 a^{3} + 10 a^{2} + 1\right)\cdot 17^{2} + \left(13 a^{3} + 6 a^{2} + 5 a + 8\right)\cdot 17^{3} + \left(2 a^{3} + 2 a^{2} + 2 a + 3\right)\cdot 17^{4} +O(17^{5})\)
|
$r_{ 11 }$ | $=$ |
\( 7 a^{3} + 13 a^{2} + 15 a + 7 + \left(15 a^{3} + 13 a^{2} + 8 a + 5\right)\cdot 17 + \left(12 a^{3} + 5 a + 3\right)\cdot 17^{2} + \left(7 a^{3} + 7 a^{2} + 4 a + 4\right)\cdot 17^{3} + \left(2 a^{3} + 11 a^{2} + 3 a + 2\right)\cdot 17^{4} +O(17^{5})\)
|
$r_{ 12 }$ | $=$ |
\( 7 a^{3} + 2 a^{2} + 10 a + 5 + \left(9 a^{3} + 12 a + 10\right)\cdot 17 + \left(11 a^{3} + 11 a^{2} + 10 a + 5\right)\cdot 17^{2} + \left(a^{3} + 11 a^{2} + 5 a + 10\right)\cdot 17^{3} + \left(5 a^{3} + 16 a^{2} + 8 a + 8\right)\cdot 17^{4} +O(17^{5})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 12 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 12 }$ | Character values | |||
$c1$ | $c2$ | $c3$ | $c4$ | |||
$1$ | $1$ | $()$ | $1$ | $1$ | $1$ | $1$ |
$1$ | $2$ | $(1,6)(2,4)(3,12)(5,9)(7,11)(8,10)$ | $-1$ | $-1$ | $-1$ | $-1$ |
$1$ | $3$ | $(1,5,7)(2,8,12)(3,4,10)(6,9,11)$ | $-\zeta_{12}^{2}$ | $-\zeta_{12}^{2}$ | $\zeta_{12}^{2} - 1$ | $\zeta_{12}^{2} - 1$ |
$1$ | $3$ | $(1,7,5)(2,12,8)(3,10,4)(6,11,9)$ | $\zeta_{12}^{2} - 1$ | $\zeta_{12}^{2} - 1$ | $-\zeta_{12}^{2}$ | $-\zeta_{12}^{2}$ |
$1$ | $4$ | $(1,10,6,8)(2,7,4,11)(3,9,12,5)$ | $\zeta_{12}^{3}$ | $-\zeta_{12}^{3}$ | $\zeta_{12}^{3}$ | $-\zeta_{12}^{3}$ |
$1$ | $4$ | $(1,8,6,10)(2,11,4,7)(3,5,12,9)$ | $-\zeta_{12}^{3}$ | $\zeta_{12}^{3}$ | $-\zeta_{12}^{3}$ | $\zeta_{12}^{3}$ |
$1$ | $6$ | $(1,11,5,6,7,9)(2,3,8,4,12,10)$ | $-\zeta_{12}^{2} + 1$ | $-\zeta_{12}^{2} + 1$ | $\zeta_{12}^{2}$ | $\zeta_{12}^{2}$ |
$1$ | $6$ | $(1,9,7,6,5,11)(2,10,12,4,8,3)$ | $\zeta_{12}^{2}$ | $\zeta_{12}^{2}$ | $-\zeta_{12}^{2} + 1$ | $-\zeta_{12}^{2} + 1$ |
$1$ | $12$ | $(1,2,9,10,7,12,6,4,5,8,11,3)$ | $\zeta_{12}$ | $-\zeta_{12}$ | $\zeta_{12}^{3} - \zeta_{12}$ | $-\zeta_{12}^{3} + \zeta_{12}$ |
$1$ | $12$ | $(1,12,11,10,5,2,6,3,7,8,9,4)$ | $\zeta_{12}^{3} - \zeta_{12}$ | $-\zeta_{12}^{3} + \zeta_{12}$ | $\zeta_{12}$ | $-\zeta_{12}$ |
$1$ | $12$ | $(1,4,9,8,7,3,6,2,5,10,11,12)$ | $-\zeta_{12}$ | $\zeta_{12}$ | $-\zeta_{12}^{3} + \zeta_{12}$ | $\zeta_{12}^{3} - \zeta_{12}$ |
$1$ | $12$ | $(1,3,11,8,5,4,6,12,7,10,9,2)$ | $-\zeta_{12}^{3} + \zeta_{12}$ | $\zeta_{12}^{3} - \zeta_{12}$ | $-\zeta_{12}$ | $\zeta_{12}$ |