Basic invariants
Dimension: | $1$ |
Group: | $C_{10}$ |
Conductor: | \(25\)\(\medspace = 5^{2} \) |
Artin field: | Galois closure of \(\Q(\zeta_{25})^+\) |
Galois orbit size: | $4$ |
Smallest permutation container: | $C_{10}$ |
Parity: | even |
Dirichlet character: | \(\chi_{25}(19,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{10} - 10x^{8} + 35x^{6} - x^{5} - 50x^{4} + 5x^{3} + 25x^{2} - 5x - 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \( x^{5} + 10x^{2} + 9 \)
Roots:
$r_{ 1 }$ | $=$ | \( 10 a^{3} + 5 a^{2} + 7 a + 5 + \left(9 a^{4} + 3 a^{3} + a^{2} + 9 a + 1\right)\cdot 11 + \left(6 a^{4} + 2 a^{3} + 3 a^{2} + a + 3\right)\cdot 11^{2} + \left(2 a^{4} + 5 a^{3} + 6 a^{2} + 9\right)\cdot 11^{3} + \left(9 a^{4} + 5 a^{3} + 4 a + 10\right)\cdot 11^{4} + \left(3 a^{4} + 2 a^{3} + 10 a^{2} + 4 a + 3\right)\cdot 11^{5} + \left(3 a^{4} + 9 a^{3} + 2 a^{2} + 8 a\right)\cdot 11^{6} +O(11^{7})\) |
$r_{ 2 }$ | $=$ | \( 8 a^{4} + 4 a^{3} + 9 a^{2} + 10 a + 2 + \left(9 a^{3} + 6 a^{2} + 1\right)\cdot 11 + \left(6 a^{4} + 6 a^{3} + 10 a^{2} + 9 a + 8\right)\cdot 11^{2} + \left(5 a^{4} + 6 a^{3} + 10 a^{2} + 6\right)\cdot 11^{3} + \left(9 a^{4} + 6 a^{3} + 10 a^{2} + 5 a + 6\right)\cdot 11^{4} + \left(4 a^{3} + 3 a^{2} + 9 a + 5\right)\cdot 11^{5} + \left(8 a^{4} + 8 a^{3} + 9 a^{2} + 2 a + 6\right)\cdot 11^{6} +O(11^{7})\) |
$r_{ 3 }$ | $=$ | \( 4 a^{4} + 4 a^{3} + a^{2} + 2 a + 2 + \left(8 a^{4} + 7 a^{3} + 3 a^{2} + 4 a\right)\cdot 11 + \left(3 a^{4} + a^{3} + 5 a^{2} + 3 a + 10\right)\cdot 11^{2} + \left(8 a^{4} + a^{3} + 4 a^{2} + 9 a + 6\right)\cdot 11^{3} + \left(8 a^{4} + 3 a^{3} + 6 a^{2} + 6 a + 7\right)\cdot 11^{4} + \left(9 a^{3} + 7 a^{2} + 10 a\right)\cdot 11^{5} + \left(9 a^{4} + 10 a^{3} + 4 a^{2} + 7 a + 10\right)\cdot 11^{6} +O(11^{7})\) |
$r_{ 4 }$ | $=$ | \( 7 a^{4} + a^{2} + 3 a + \left(10 a^{4} + 2 a^{3} + 9 a^{2} + 2 a + 1\right)\cdot 11 + \left(7 a^{4} + 10 a^{3} + 10 a^{2} + 4 a + 6\right)\cdot 11^{2} + \left(4 a^{4} + 6 a^{3} + 7 a + 8\right)\cdot 11^{3} + \left(7 a^{4} + 7 a^{3} + 9 a^{2} + 1\right)\cdot 11^{4} + \left(9 a^{3} + 2 a^{2} + 7 a + 3\right)\cdot 11^{5} + \left(2 a^{4} + 5 a^{3} + 10 a^{2} + 6 a + 2\right)\cdot 11^{6} +O(11^{7})\) |
$r_{ 5 }$ | $=$ | \( 5 a^{4} + a^{3} + 2 a^{2} + 3 a + 6 + \left(4 a^{4} + 10 a^{3} + 3 a^{2} + 5 a + 5\right)\cdot 11 + \left(6 a^{4} + 5 a^{3} + 3 a^{2} + 7 a + 2\right)\cdot 11^{2} + \left(10 a^{4} + 2 a^{3} + 9 a^{2} + 3 a + 4\right)\cdot 11^{3} + \left(3 a^{3} + 4 a^{2} + 4 a + 8\right)\cdot 11^{4} + \left(6 a^{4} + 5 a^{3} + 10 a^{2} + 9\right)\cdot 11^{5} + \left(8 a^{4} + 8 a^{3} + 9 a^{2} + 2 a + 6\right)\cdot 11^{6} +O(11^{7})\) |
$r_{ 6 }$ | $=$ | \( a^{4} + a^{3} + 9 a^{2} + 10 a + 6 + \left(6 a^{4} + 3 a^{3} + 6 a^{2} + 8 a + 7\right)\cdot 11 + \left(5 a^{4} + 7 a^{3} + 3 a^{2} + 10 a + 10\right)\cdot 11^{2} + \left(6 a^{4} + 6 a^{3} + a + 6\right)\cdot 11^{3} + \left(3 a^{4} + 3 a^{3} + 7 a^{2} + 3 a + 10\right)\cdot 11^{4} + \left(a^{4} + 9 a^{3} + 10 a^{2} + 3 a\right)\cdot 11^{5} + \left(a^{4} + 3 a^{3} + 10 a^{2} + 7 a + 1\right)\cdot 11^{6} +O(11^{7})\) |
$r_{ 7 }$ | $=$ | \( 6 a^{4} + 7 a^{3} + 2 a^{2} + 7 a + 9 + \left(9 a^{3} + 5 a^{2} + 2\right)\cdot 11 + \left(8 a^{4} + a^{3} + 10 a^{2} + 5 a\right)\cdot 11^{2} + \left(6 a^{4} + 6 a^{3} + a + 4\right)\cdot 11^{3} + \left(6 a^{4} + 2 a^{3} + a^{2} + 6 a + 4\right)\cdot 11^{4} + \left(10 a^{4} + 6 a^{3} + 2 a^{2} + 10 a + 4\right)\cdot 11^{5} + \left(9 a^{4} + 9 a^{3} + 5 a^{2} + 7 a + 2\right)\cdot 11^{6} +O(11^{7})\) |
$r_{ 8 }$ | $=$ | \( 5 a^{3} + 10 a^{2} + 5 a + 8 + \left(2 a^{3} + 4 a^{2} + a + 3\right)\cdot 11 + \left(a^{4} + 3 a^{3} + 10 a^{2} + 5 a + 8\right)\cdot 11^{2} + \left(6 a^{4} + 6 a^{2} + 3 a + 1\right)\cdot 11^{3} + \left(9 a^{4} + a^{3} + 4 a^{2} + 9 a + 6\right)\cdot 11^{4} + \left(6 a^{4} + 4 a^{3} + 7 a^{2} + 5 a + 2\right)\cdot 11^{5} + \left(6 a^{4} + 6 a^{3} + 3 a^{2} + 5\right)\cdot 11^{6} +O(11^{7})\) |
$r_{ 9 }$ | $=$ | \( 3 a^{4} + 9 a^{3} + 6 a^{2} + 3 a + 10 + \left(10 a^{4} + 9 a^{3} + 9 a^{2} + 3\right)\cdot 11 + \left(5 a^{3} + 9 a^{2} + 10 a + 2\right)\cdot 11^{2} + \left(9 a^{2} + 9 a + 3\right)\cdot 11^{3} + \left(5 a^{2} + 5 a\right)\cdot 11^{4} + \left(6 a^{4} + a^{3} + a^{2} + 7 a + 6\right)\cdot 11^{5} + \left(9 a^{4} + 7 a^{2} + 10 a\right)\cdot 11^{6} +O(11^{7})\) |
$r_{ 10 }$ | $=$ | \( 10 a^{4} + 3 a^{3} + 10 a^{2} + 5 a + 7 + \left(4 a^{4} + 8 a^{3} + 4 a^{2} + 10 a + 5\right)\cdot 11 + \left(8 a^{4} + 9 a^{3} + 9 a^{2} + 8 a + 3\right)\cdot 11^{2} + \left(3 a^{4} + 7 a^{3} + 4 a^{2} + 5 a + 3\right)\cdot 11^{3} + \left(10 a^{4} + 10 a^{3} + 4 a^{2} + 9 a + 9\right)\cdot 11^{4} + \left(6 a^{4} + 2 a^{3} + 9 a^{2} + 6 a + 6\right)\cdot 11^{5} + \left(7 a^{4} + 3 a^{3} + a^{2} + 8\right)\cdot 11^{6} +O(11^{7})\) |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,9)(2,3)(4,6)(5,8)(7,10)$ | $-1$ |
$1$ | $5$ | $(1,7,4,3,5)(2,8,9,10,6)$ | $\zeta_{5}^{3}$ |
$1$ | $5$ | $(1,4,5,7,3)(2,9,6,8,10)$ | $\zeta_{5}$ |
$1$ | $5$ | $(1,3,7,5,4)(2,10,8,6,9)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
$1$ | $5$ | $(1,5,3,4,7)(2,6,10,9,8)$ | $\zeta_{5}^{2}$ |
$1$ | $10$ | $(1,10,4,2,5,9,7,6,3,8)$ | $-\zeta_{5}^{3}$ |
$1$ | $10$ | $(1,2,7,8,4,9,3,10,5,6)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ |
$1$ | $10$ | $(1,6,5,10,3,9,4,8,7,2)$ | $-\zeta_{5}$ |
$1$ | $10$ | $(1,8,3,6,7,9,5,2,4,10)$ | $-\zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.