Basic invariants
Dimension: | $1$ |
Group: | $C_{10}$ |
Conductor: | \(77\)\(\medspace = 7 \cdot 11 \) |
Artin field: | Galois closure of 10.0.3602729712967.1 |
Galois orbit size: | $4$ |
Smallest permutation container: | $C_{10}$ |
Parity: | odd |
Dirichlet character: | \(\chi_{77}(69,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{10} - x^{9} + 14x^{8} - 7x^{7} + 85x^{6} - 29x^{5} + 218x^{4} - 8x^{3} + 216x^{2} - 48x + 32 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: \( x^{5} + 3x + 27 \)
Roots:
$r_{ 1 }$ | $=$ | \( 2 a^{4} + 20 a^{3} + 12 a^{2} + 18 a + 18 + \left(15 a^{4} + 26 a^{3} + 12 a^{2} + 16 a\right)\cdot 29 + \left(9 a^{4} + 28 a^{3} + 13 a^{2} + 8 a + 23\right)\cdot 29^{2} + \left(19 a^{4} + a^{3} + 23 a^{2} + 24 a + 2\right)\cdot 29^{3} + \left(a^{4} + 20 a^{3} + 11 a^{2} + 24 a + 5\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 2 }$ | $=$ | \( 4 a^{4} + 6 a^{3} + 14 a^{2} + 19 a + 14 + \left(9 a^{4} + 15 a^{3} + 26 a^{2} + 17 a + 28\right)\cdot 29 + \left(4 a^{4} + 27 a^{3} + 11 a^{2} + 6 a + 15\right)\cdot 29^{2} + \left(26 a^{4} + 5 a^{3} + 17 a^{2} + 8 a + 13\right)\cdot 29^{3} + \left(17 a^{4} + 24 a^{3} + 7 a + 24\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 3 }$ | $=$ | \( 5 a^{4} + 10 a^{3} + 13 a^{2} + 24 a + 28 + \left(18 a^{4} + 25 a^{3} + 12 a^{2} + 23 a + 26\right)\cdot 29 + \left(22 a^{4} + 12 a^{3} + 11 a^{2} + 23 a + 1\right)\cdot 29^{2} + \left(14 a^{4} + 7 a^{3} + 15 a^{2} + 6 a + 21\right)\cdot 29^{3} + \left(5 a^{3} + 25 a^{2} + 7 a + 11\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 4 }$ | $=$ | \( 6 a^{4} + 5 a^{3} + 21 a^{2} + 2 a + 13 + \left(12 a^{4} + 12 a^{3} + 22 a^{2} + 8 a + 18\right)\cdot 29 + \left(11 a^{4} + 9 a^{3} + 27 a^{2} + 3 a + 15\right)\cdot 29^{2} + \left(23 a^{4} + 6 a^{3} + 3 a^{2} + 2 a + 18\right)\cdot 29^{3} + \left(22 a^{4} + 17 a^{3} + 7 a + 24\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 5 }$ | $=$ | \( 11 a^{4} + 4 a^{3} + 18 a^{2} + 21 a + 28 + \left(12 a^{4} + 2 a^{3} + 14 a^{2} + 11 a + 28\right)\cdot 29 + \left(19 a^{4} + 18 a^{3} + 19 a^{2} + 10 a + 11\right)\cdot 29^{2} + \left(24 a^{3} + 7 a^{2} + 14 a + 4\right)\cdot 29^{3} + \left(18 a^{4} + 12 a^{3} + 19 a + 21\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 6 }$ | $=$ | \( 12 a^{4} + 18 a^{3} + 13 a^{2} + 28 a + 13 + \left(19 a^{4} + 4 a^{3} + 22 a^{2} + 14 a + 28\right)\cdot 29 + \left(25 a^{4} + 26 a^{3} + 18 a^{2} + 8 a + 26\right)\cdot 29^{2} + \left(26 a^{3} + 23 a^{2} + 8 a + 4\right)\cdot 29^{3} + \left(5 a^{4} + 21 a^{3} + 4 a^{2} + 20 a + 13\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 7 }$ | $=$ | \( 15 a^{4} + a^{3} + 10 a^{2} + 14 a + 26 + \left(15 a^{4} + 27 a^{3} + 11 a^{2} + 23 a + 24\right)\cdot 29 + \left(19 a^{4} + 21 a^{3} + a^{2} + 6 a + 23\right)\cdot 29^{2} + \left(28 a^{4} + 6 a^{3} + 19 a^{2} + 22 a + 7\right)\cdot 29^{3} + \left(6 a^{4} + 21 a^{3} + 11 a^{2} + 14 a + 6\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 8 }$ | $=$ | \( 18 a^{4} + 15 a^{3} + 5 a^{2} + 6 a + 10 + \left(24 a^{4} + 26 a^{3} + 26 a^{2} + 20 a\right)\cdot 29 + \left(12 a^{4} + 20 a^{3} + 4 a^{2} + 23 a + 2\right)\cdot 29^{2} + \left(8 a^{4} + 26 a^{3} + 13 a^{2} + 17 a + 23\right)\cdot 29^{3} + \left(26 a^{4} + 10 a^{3} + 7 a + 17\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 9 }$ | $=$ | \( 20 a^{4} + 26 a^{3} + 4 a^{2} + 6 a + 6 + \left(8 a^{4} + 6 a^{3} + 26 a^{2} + 19 a + 4\right)\cdot 29 + \left(15 a^{4} + a^{2} + 14 a + 19\right)\cdot 29^{2} + \left(26 a^{4} + 26 a^{3} + 5 a^{2} + 5 a + 8\right)\cdot 29^{3} + \left(2 a^{4} + 24 a^{3} + 15 a^{2} + 3 a + 23\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 10 }$ | $=$ | \( 23 a^{4} + 11 a^{3} + 6 a^{2} + 7 a + 19 + \left(9 a^{4} + 27 a^{3} + 28 a^{2} + 18 a + 12\right)\cdot 29 + \left(4 a^{4} + 7 a^{3} + 4 a^{2} + 9 a + 4\right)\cdot 29^{2} + \left(25 a^{4} + 12 a^{3} + 16 a^{2} + 6 a + 11\right)\cdot 29^{3} + \left(13 a^{4} + 15 a^{3} + 16 a^{2} + 4 a + 26\right)\cdot 29^{4} +O(29^{5})\) |
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 value |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,9)(2,6)(3,7)(4,8)(5,10)$ | $-1$ |
$1$ | $5$ | $(1,6,5,8,7)(2,10,4,3,9)$ | $\zeta_{5}$ |
$1$ | $5$ | $(1,5,7,6,8)(2,4,9,10,3)$ | $\zeta_{5}^{2}$ |
$1$ | $5$ | $(1,8,6,7,5)(2,3,10,9,4)$ | $\zeta_{5}^{3}$ |
$1$ | $5$ | $(1,7,8,5,6)(2,9,3,4,10)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
$1$ | $10$ | $(1,4,6,3,5,9,8,2,7,10)$ | $-\zeta_{5}^{3}$ |
$1$ | $10$ | $(1,3,8,10,6,9,7,4,5,2)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ |
$1$ | $10$ | $(1,2,5,4,7,9,6,10,8,3)$ | $-\zeta_{5}$ |
$1$ | $10$ | $(1,10,7,2,8,9,5,3,6,4)$ | $-\zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.