Basic invariants
Dimension: | $1$ |
Group: | $C_{10}$ |
Conductor: | \(2600\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 13 \) |
Artin field: | Galois closure of 10.0.1856465000000000000000.3 |
Galois orbit size: | $4$ |
Smallest permutation container: | $C_{10}$ |
Parity: | odd |
Dirichlet character: | \(\chi_{2600}(571,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{10} + 110 x^{8} - 10 x^{7} + 5840 x^{6} + 358 x^{5} + 180785 x^{4} + 39440 x^{3} + 3124530 x^{2} + \cdots + 23284457 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \( x^{5} + x + 14 \)
Roots:
$r_{ 1 }$ | $=$ | \( a^{4} + 13 a^{3} + 4 a^{2} + 7 a + 6 + \left(12 a^{3} + 2 a^{2} + 2 a + 15\right)\cdot 17 + \left(8 a^{3} + 6 a^{2} + 3 a + 8\right)\cdot 17^{2} + \left(14 a^{4} + 11 a^{3} + 8 a^{2} + 7 a + 3\right)\cdot 17^{3} + \left(12 a^{4} + 10 a^{3} + 14 a^{2} + 2 a + 5\right)\cdot 17^{4} + \left(5 a^{4} + 13 a^{3} + 6 a^{2} + 5 a + 7\right)\cdot 17^{5} +O(17^{6})\) |
$r_{ 2 }$ | $=$ | \( a^{4} + 13 a^{3} + 4 a^{2} + 7 a + 16 + \left(12 a^{3} + 2 a^{2} + 2 a + 11\right)\cdot 17 + \left(8 a^{3} + 6 a^{2} + 3 a + 4\right)\cdot 17^{2} + \left(14 a^{4} + 11 a^{3} + 8 a^{2} + 7 a + 5\right)\cdot 17^{3} + \left(12 a^{4} + 10 a^{3} + 14 a^{2} + 2 a + 15\right)\cdot 17^{4} + \left(5 a^{4} + 13 a^{3} + 6 a^{2} + 5 a + 1\right)\cdot 17^{5} +O(17^{6})\) |
$r_{ 3 }$ | $=$ | \( 11 a^{4} + 9 a^{2} + 12 a + 7 + \left(6 a^{4} + 10 a^{3} + 3 a^{2} + 7 a + 10\right)\cdot 17 + \left(5 a^{4} + 11 a^{3} + 4 a^{2} + 8 a + 5\right)\cdot 17^{2} + \left(a^{4} + 13 a^{3} + 12 a^{2} + 5 a + 5\right)\cdot 17^{3} + \left(7 a^{4} + 12 a^{3} + 12 a^{2} + 7 a + 7\right)\cdot 17^{4} + \left(6 a^{4} + 13 a^{3} + 5 a^{2} + 12 a + 2\right)\cdot 17^{5} +O(17^{6})\) |
$r_{ 4 }$ | $=$ | \( 11 a^{4} + 9 a^{2} + 12 a + 14 + \left(6 a^{4} + 10 a^{3} + 3 a^{2} + 7 a + 13\right)\cdot 17 + \left(5 a^{4} + 11 a^{3} + 4 a^{2} + 8 a + 9\right)\cdot 17^{2} + \left(a^{4} + 13 a^{3} + 12 a^{2} + 5 a + 3\right)\cdot 17^{3} + \left(7 a^{4} + 12 a^{3} + 12 a^{2} + 7 a + 14\right)\cdot 17^{4} + \left(6 a^{4} + 13 a^{3} + 5 a^{2} + 12 a + 7\right)\cdot 17^{5} +O(17^{6})\) |
$r_{ 5 }$ | $=$ | \( 11 a^{4} + 7 a^{3} + 10 a^{2} + 16 a + 7 + \left(8 a^{4} + 4 a^{3} + 15 a^{2} + 14 a + 15\right)\cdot 17 + \left(13 a^{4} + 14 a^{3} + 16 a^{2} + 11 a + 1\right)\cdot 17^{2} + \left(7 a^{4} + 3 a^{3} + 8 a^{2} + 10 a + 7\right)\cdot 17^{3} + \left(8 a^{4} + 10 a^{3} + 14 a^{2} + 5 a + 8\right)\cdot 17^{4} + \left(16 a^{4} + 15 a^{3} + 10 a^{2} + 10 a + 10\right)\cdot 17^{5} +O(17^{6})\) |
$r_{ 6 }$ | $=$ | \( 11 a^{4} + 7 a^{3} + 10 a^{2} + 16 a + 14 + \left(8 a^{4} + 4 a^{3} + 15 a^{2} + 14 a + 1\right)\cdot 17 + \left(13 a^{4} + 14 a^{3} + 16 a^{2} + 11 a + 6\right)\cdot 17^{2} + \left(7 a^{4} + 3 a^{3} + 8 a^{2} + 10 a + 5\right)\cdot 17^{3} + \left(8 a^{4} + 10 a^{3} + 14 a^{2} + 5 a + 15\right)\cdot 17^{4} + \left(16 a^{4} + 15 a^{3} + 10 a^{2} + 10 a + 15\right)\cdot 17^{5} +O(17^{6})\) |
$r_{ 7 }$ | $=$ | \( 14 a^{4} + 7 a^{3} + a^{2} + 3 a + 6 + \left(4 a^{4} + 9 a^{3} + 3 a^{2} + 16 a + 12\right)\cdot 17 + \left(3 a^{4} + 10 a^{3} + 7 a^{2} + 7 a + 10\right)\cdot 17^{2} + \left(11 a^{4} + 12 a^{3} + 15 a^{2} + 4 a + 16\right)\cdot 17^{3} + \left(a^{4} + 3 a^{3} + 14 a^{2} + a + 2\right)\cdot 17^{4} + \left(2 a^{4} + 7 a^{3} + 3 a + 9\right)\cdot 17^{5} +O(17^{6})\) |
$r_{ 8 }$ | $=$ | \( 14 a^{4} + 7 a^{3} + a^{2} + 3 a + 13 + \left(4 a^{4} + 9 a^{3} + 3 a^{2} + 16 a + 15\right)\cdot 17 + \left(3 a^{4} + 10 a^{3} + 7 a^{2} + 7 a + 14\right)\cdot 17^{2} + \left(11 a^{4} + 12 a^{3} + 15 a^{2} + 4 a + 14\right)\cdot 17^{3} + \left(a^{4} + 3 a^{3} + 14 a^{2} + a + 9\right)\cdot 17^{4} + \left(2 a^{4} + 7 a^{3} + 3 a + 14\right)\cdot 17^{5} +O(17^{6})\) |
$r_{ 9 }$ | $=$ | \( 14 a^{4} + 7 a^{3} + 10 a^{2} + 13 a + 6 + \left(13 a^{4} + 14 a^{3} + 9 a^{2} + 9 a + 9\right)\cdot 17 + \left(11 a^{4} + 5 a^{3} + 16 a^{2} + 2 a\right)\cdot 17^{2} + \left(16 a^{4} + 9 a^{3} + 5 a^{2} + 6 a + 4\right)\cdot 17^{3} + \left(3 a^{4} + 13 a^{3} + 11 a^{2} + 8\right)\cdot 17^{4} + \left(3 a^{4} + 9 a^{2} + 3 a + 13\right)\cdot 17^{5} +O(17^{6})\) |
$r_{ 10 }$ | $=$ | \( 14 a^{4} + 7 a^{3} + 10 a^{2} + 13 a + 13 + \left(13 a^{4} + 14 a^{3} + 9 a^{2} + 9 a + 12\right)\cdot 17 + \left(11 a^{4} + 5 a^{3} + 16 a^{2} + 2 a + 4\right)\cdot 17^{2} + \left(16 a^{4} + 9 a^{3} + 5 a^{2} + 6 a + 2\right)\cdot 17^{3} + \left(3 a^{4} + 13 a^{3} + 11 a^{2} + 15\right)\cdot 17^{4} + \left(3 a^{4} + 9 a^{2} + 3 a + 1\right)\cdot 17^{5} +O(17^{6})\) |
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 value |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,2)(3,4)(5,6)(7,8)(9,10)$ | $-1$ |
$1$ | $5$ | $(1,8,6,4,10)(2,7,5,3,9)$ | $\zeta_{5}^{2}$ |
$1$ | $5$ | $(1,6,10,8,4)(2,5,9,7,3)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
$1$ | $5$ | $(1,4,8,10,6)(2,3,7,9,5)$ | $\zeta_{5}$ |
$1$ | $5$ | $(1,10,4,6,8)(2,9,3,5,7)$ | $\zeta_{5}^{3}$ |
$1$ | $10$ | $(1,3,8,9,6,2,4,7,10,5)$ | $-\zeta_{5}$ |
$1$ | $10$ | $(1,9,4,5,8,2,10,3,6,7)$ | $-\zeta_{5}^{3}$ |
$1$ | $10$ | $(1,7,6,3,10,2,8,5,4,9)$ | $-\zeta_{5}^{2}$ |
$1$ | $10$ | $(1,5,10,7,4,2,6,9,8,3)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.