Basic invariants
Dimension: | $1$ |
Group: | $C_{10}$ |
Conductor: | \(33\)\(\medspace = 3 \cdot 11 \) |
Artin number field: | Galois closure of \(\Q(\zeta_{33})^+\) |
Galois orbit size: | $4$ |
Smallest permutation container: | $C_{10}$ |
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 $\Q_{ 17 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$:
\( x^{5} + x + 14 \)
Roots:
$r_{ 1 }$ | $=$ | \( 14 a^{3} + 16 a^{2} + 14 a + 15 + \left(4 a^{4} + 11 a^{3} + 2 a^{2} + 5 a + 11\right)\cdot 17 + \left(9 a^{4} + 5 a^{3} + 3 a^{2} + 4 a + 1\right)\cdot 17^{2} + \left(14 a^{4} + 6 a^{3} + 4 a^{2} + 13 a + 14\right)\cdot 17^{3} + \left(15 a^{4} + 4 a^{3} + 11 a^{2} + 11 a + 1\right)\cdot 17^{4} +O(17^{5})\) |
$r_{ 2 }$ | $=$ | \( 2 a^{4} + a^{3} + 4 a^{2} + 16 a + 3 + \left(7 a^{4} + 11 a^{3} + 4 a^{2} + 11 a + 4\right)\cdot 17 + \left(10 a^{4} + 15 a^{3} + 11 a^{2} + 14 a + 16\right)\cdot 17^{2} + \left(2 a^{4} + 4 a^{3} + 9 a^{2} + 8 a + 7\right)\cdot 17^{3} + \left(6 a^{4} + 12 a^{3} + 9 a^{2} + 10 a + 14\right)\cdot 17^{4} +O(17^{5})\) |
$r_{ 3 }$ | $=$ | \( 2 a^{4} + a^{3} + 12 a^{2} + 9 a + 14 + \left(4 a^{4} + 5 a^{3} + 11 a^{2} + 8 a + 4\right)\cdot 17 + \left(14 a^{4} + 3 a^{3} + 11 a^{2} + 5 a + 10\right)\cdot 17^{2} + \left(5 a^{4} + 3 a^{3} + 2 a + 5\right)\cdot 17^{3} + \left(9 a^{4} + 8 a^{3} + 9 a^{2} + 15 a + 1\right)\cdot 17^{4} +O(17^{5})\) |
$r_{ 4 }$ | $=$ | \( 3 a^{4} + 13 a^{3} + 5 a^{2} + a + 14 + \left(8 a^{3} + 12 a^{2} + 8 a + 8\right)\cdot 17 + \left(15 a^{4} + 15 a^{3} + 15 a^{2} + 11 a + 16\right)\cdot 17^{2} + \left(6 a^{4} + 13 a^{3} + 14 a^{2} + 14 a + 7\right)\cdot 17^{3} + \left(11 a^{4} + 6 a^{3} + a^{2} + 10 a + 8\right)\cdot 17^{4} +O(17^{5})\) |
$r_{ 5 }$ | $=$ | \( 5 a^{4} + 15 a^{2} + 16 a + 13 + \left(6 a^{4} + 10 a^{3} + 6 a^{2} + 4 a + 16\right)\cdot 17 + \left(3 a^{4} + a^{3} + 13 a^{2} + 12 a + 4\right)\cdot 17^{2} + \left(10 a^{4} + 3 a^{3} + 10 a^{2} + 5 a + 2\right)\cdot 17^{3} + \left(16 a^{4} + 11 a^{3} + 7 a^{2} + 14 a + 7\right)\cdot 17^{4} +O(17^{5})\) |
$r_{ 6 }$ | $=$ | \( 5 a^{4} + 6 a^{3} + 3 a^{2} + 7 a + 2 + \left(14 a^{4} + 14 a^{3} + 7 a^{2} + 14 a + 3\right)\cdot 17 + \left(a^{4} + 3 a^{3} + 15 a^{2} + 12 a + 16\right)\cdot 17^{2} + \left(5 a^{4} + 11 a^{3} + 3 a^{2} + 7 a + 9\right)\cdot 17^{3} + \left(12 a^{4} + 5 a^{3} + 6 a^{2} + 7 a + 12\right)\cdot 17^{4} +O(17^{5})\) |
$r_{ 7 }$ | $=$ | \( 7 a^{4} + 6 a^{2} + 13 a + 7 + \left(8 a^{4} + 5 a^{3} + 7 a^{2} + 10 a + 15\right)\cdot 17 + \left(14 a^{4} + 10 a^{3} + 5 a^{2} + 7 a + 5\right)\cdot 17^{2} + \left(4 a^{4} + 14 a^{3} + a^{2} + 6 a + 6\right)\cdot 17^{3} + \left(5 a^{4} + 4 a^{3} + 5 a^{2} + 10 a + 10\right)\cdot 17^{4} +O(17^{5})\) |
$r_{ 8 }$ | $=$ | \( 7 a^{4} + 11 a^{3} + 3 a^{2} + 9 a + 1 + \left(10 a^{4} + 13 a^{3} + 3 a^{2} + 15 a + 3\right)\cdot 17 + \left(11 a^{4} + 9 a^{3} + 12 a^{2} + 8 a + 8\right)\cdot 17^{2} + \left(14 a^{4} + 12 a^{3} + 16 a^{2} + 13 a + 2\right)\cdot 17^{3} + \left(15 a^{4} + 4 a^{3} + 5 a^{2} + a + 3\right)\cdot 17^{4} +O(17^{5})\) |
$r_{ 9 }$ | $=$ | \( 10 a^{4} + 8 a^{3} + 15 a^{2} + a + \left(11 a^{4} + 14 a^{3} + 4 a^{2} + 10 a + 4\right)\cdot 17 + \left(6 a^{4} + 9 a^{3} + 8 a^{2} + 12 a + 4\right)\cdot 17^{2} + \left(9 a^{4} + 9 a^{3} + 14 a^{2} + 16 a + 15\right)\cdot 17^{3} + \left(2 a^{4} + a^{3} + 15 a^{2} + a + 12\right)\cdot 17^{4} +O(17^{5})\) |
$r_{ 10 }$ | $=$ | \( 10 a^{4} + 14 a^{3} + 6 a^{2} + 16 a + \left(a^{4} + 7 a^{3} + 7 a^{2} + 11 a + 13\right)\cdot 17 + \left(15 a^{4} + 9 a^{3} + 5 a^{2} + 11 a\right)\cdot 17^{2} + \left(10 a^{4} + 5 a^{3} + 8 a^{2} + 12 a + 13\right)\cdot 17^{3} + \left(6 a^{4} + 8 a^{3} + 12 a^{2} + 12\right)\cdot 17^{4} +O(17^{5})\) |
Generators of the action on the roots $r_1, \ldots, r_{ 10 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 10 }$ | Character values | |||
$c1$ | $c2$ | $c3$ | $c4$ | |||
$1$ | $1$ | $()$ | $1$ | $1$ | $1$ | $1$ |
$1$ | $2$ | $(1,3)(2,10)(4,5)(6,9)(7,8)$ | $-1$ | $-1$ | $-1$ | $-1$ |
$1$ | $5$ | $(1,2,6,4,7)(3,10,9,5,8)$ | $\zeta_{5}$ | $\zeta_{5}^{2}$ | $\zeta_{5}^{3}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
$1$ | $5$ | $(1,6,7,2,4)(3,9,8,10,5)$ | $\zeta_{5}^{2}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}$ | $\zeta_{5}^{3}$ |
$1$ | $5$ | $(1,4,2,7,6)(3,5,10,8,9)$ | $\zeta_{5}^{3}$ | $\zeta_{5}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}^{2}$ |
$1$ | $5$ | $(1,7,4,6,2)(3,8,5,9,10)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}^{3}$ | $\zeta_{5}^{2}$ | $\zeta_{5}$ |
$1$ | $10$ | $(1,5,2,8,6,3,4,10,7,9)$ | $-\zeta_{5}^{3}$ | $-\zeta_{5}$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ | $-\zeta_{5}^{2}$ |
$1$ | $10$ | $(1,8,4,9,2,3,7,5,6,10)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ | $-\zeta_{5}^{3}$ | $-\zeta_{5}^{2}$ | $-\zeta_{5}$ |
$1$ | $10$ | $(1,10,6,5,7,3,2,9,4,8)$ | $-\zeta_{5}$ | $-\zeta_{5}^{2}$ | $-\zeta_{5}^{3}$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ |
$1$ | $10$ | $(1,9,7,10,4,3,6,8,2,5)$ | $-\zeta_{5}^{2}$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ | $-\zeta_{5}$ | $-\zeta_{5}^{3}$ |