Basic invariants
Dimension: | $2$ |
Group: | $D_{10}$ |
Conductor: | \(11163\)\(\medspace = 3 \cdot 61^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 10.2.947225833519565421.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $D_{10}$ |
Parity: | odd |
Determinant: | 1.3.2t1.a.a |
Projective image: | $D_5$ |
Projective stem field: | Galois closure of 5.1.124612569.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{10} - 3x^{9} + x^{8} + 48x^{7} - 74x^{6} + 378x^{5} + 485x^{4} - 1887x^{3} + 3077x^{2} + 4866x - 13372 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \( x^{5} + 4x + 11 \)
Roots:
$r_{ 1 }$ | $=$ | \( 6 a^{3} + 6 a^{2} + 8 a + \left(7 a^{4} + 2 a^{3} + a^{2} + 5 a + 5\right)\cdot 13 + \left(8 a^{4} + 12 a^{3} + 10 a + 2\right)\cdot 13^{2} + \left(9 a^{4} + 12 a^{3} + 4 a^{2} + 10 a + 4\right)\cdot 13^{3} + \left(11 a^{4} + a^{3} + 2 a^{2} + 3 a + 11\right)\cdot 13^{4} + \left(12 a^{4} + 6 a^{3} + 2 a^{2} + 7 a + 6\right)\cdot 13^{5} + \left(a^{4} + 2 a^{3} + 7 a^{2} + 12 a + 7\right)\cdot 13^{6} + \left(2 a^{4} + 6 a^{3} + 10 a^{2} + 6 a + 1\right)\cdot 13^{7} + \left(11 a^{4} + 4 a^{3} + 10 a^{2} + 10 a + 1\right)\cdot 13^{8} + \left(11 a^{4} + 3 a^{3} + 2 a^{2} + 11\right)\cdot 13^{9} +O(13^{10})\) |
$r_{ 2 }$ | $=$ | \( 5 a^{4} + 4 a^{3} + 8 a^{2} + 1 + \left(2 a^{4} + a^{3} + 8 a^{2} + 4 a + 12\right)\cdot 13 + \left(7 a^{4} + 7 a^{3} + 2 a^{2} + 2 a + 8\right)\cdot 13^{2} + \left(12 a^{4} + 10 a^{3} + 11 a^{2} + 2 a + 9\right)\cdot 13^{3} + \left(2 a^{4} + 9 a^{3} + 6 a^{2} + 12 a + 9\right)\cdot 13^{4} + \left(3 a^{3} + 3 a^{2} + 5 a + 11\right)\cdot 13^{5} + \left(2 a^{4} + 5 a^{3} + 6 a^{2} + 10 a + 7\right)\cdot 13^{6} + \left(12 a^{4} + 11 a^{3} + 11 a^{2} + 11 a + 2\right)\cdot 13^{7} + \left(2 a^{4} + 8 a^{3} + 7 a^{2} + 3 a + 10\right)\cdot 13^{8} + \left(11 a^{4} + 12 a^{3} + 5 a^{2} + 9 a + 10\right)\cdot 13^{9} +O(13^{10})\) |
$r_{ 3 }$ | $=$ | \( 7 a^{4} + a^{3} + 11 a^{2} + 4 a + 10 + \left(2 a^{4} + 6 a^{3} + 11 a^{2} + 7 a + 4\right)\cdot 13 + \left(10 a^{4} + 10 a^{3} + 10 a^{2} + 4 a\right)\cdot 13^{2} + \left(8 a^{4} + 4 a^{3} + 10 a^{2} + 3 a + 8\right)\cdot 13^{3} + \left(11 a^{4} + 10 a^{3} + 7 a^{2} + 3 a + 11\right)\cdot 13^{4} + \left(11 a^{4} + 2 a^{3} + 5 a^{2} + 10 a + 4\right)\cdot 13^{5} + \left(4 a^{4} + 9 a^{3} + 4 a^{2} + 8 a + 9\right)\cdot 13^{6} + \left(7 a^{4} + 8 a^{3} + 10 a^{2} + 11 a + 10\right)\cdot 13^{7} + \left(11 a^{4} + 6 a^{3} + 8 a^{2} + a + 11\right)\cdot 13^{8} + \left(2 a^{4} + 6 a^{3} + 6 a^{2} + 11 a + 4\right)\cdot 13^{9} +O(13^{10})\) |
$r_{ 4 }$ | $=$ | \( 4 a^{4} + 9 a^{3} + 9 a^{2} + 4 a + 3 + \left(a^{4} + 6 a^{3} + 5 a^{2} + 11 a + 11\right)\cdot 13 + \left(4 a^{4} + 6 a^{3} + 9 a^{2} + 5 a + 6\right)\cdot 13^{2} + \left(3 a^{4} + 6 a^{3} + 10 a^{2} + a + 3\right)\cdot 13^{3} + \left(7 a^{4} + a^{3} + 4 a + 5\right)\cdot 13^{4} + \left(10 a^{4} + 12 a^{3} + 4 a^{2} + 8 a + 8\right)\cdot 13^{5} + \left(7 a^{4} + 5 a^{3} + 2 a^{2} + 4 a + 5\right)\cdot 13^{6} + \left(8 a^{4} + 2 a^{3} + 8 a^{2} + 5 a + 9\right)\cdot 13^{7} + \left(7 a^{4} + 6 a^{3} + 6 a + 9\right)\cdot 13^{8} + \left(4 a^{4} + 4 a^{3} + 2 a^{2} + 6 a + 7\right)\cdot 13^{9} +O(13^{10})\) |
$r_{ 5 }$ | $=$ | \( 2 a^{4} + 6 a^{3} + 4 a^{2} + a + 7 + \left(10 a^{4} + 6 a^{3} + 6 a^{2} + 3 a\right)\cdot 13 + \left(3 a^{4} + 3 a^{3} + 12 a^{2} + 3 a + 11\right)\cdot 13^{2} + \left(10 a^{4} + 7 a^{3} + 10 a^{2} + 4 a + 12\right)\cdot 13^{3} + \left(8 a^{4} + 8 a^{3} + 2 a^{2} + 10 a + 4\right)\cdot 13^{4} + \left(12 a^{4} + 3 a^{3} + 3 a^{2} + 10 a + 7\right)\cdot 13^{5} + \left(3 a^{3} + 4 a + 4\right)\cdot 13^{6} + \left(4 a^{4} + 8 a^{3} + 2 a^{2} + 4 a + 5\right)\cdot 13^{7} + \left(6 a^{3} + 9 a^{2} + 7 a + 4\right)\cdot 13^{8} + \left(4 a^{4} + 3 a^{3} + 11 a^{2} + a + 3\right)\cdot 13^{9} +O(13^{10})\) |
$r_{ 6 }$ | $=$ | \( 5 a^{4} + 6 a^{3} + a^{2} + 9 a + 3 + \left(4 a^{4} + 8 a^{3} + 4 a + 12\right)\cdot 13 + \left(2 a^{4} + 11 a^{3} + 5 a^{2} + a + 5\right)\cdot 13^{2} + \left(9 a^{4} + 2 a^{3} + 12 a^{2} + 10\right)\cdot 13^{3} + \left(7 a^{4} + 2 a^{3} + 4 a^{2} + 7 a + 8\right)\cdot 13^{4} + \left(6 a^{4} + 6 a^{3} + 12 a^{2} + 8 a + 12\right)\cdot 13^{5} + \left(10 a^{4} + a^{3} + 4 a^{2} + 5 a\right)\cdot 13^{6} + \left(9 a^{4} + 5 a^{3} + a^{2} + 7 a\right)\cdot 13^{7} + \left(8 a^{4} + 4 a^{3} + 12 a^{2} + a + 9\right)\cdot 13^{8} + \left(5 a^{4} + 9 a^{2} + 6 a + 1\right)\cdot 13^{9} +O(13^{10})\) |
$r_{ 7 }$ | $=$ | \( 10 a^{2} + 5 a + \left(12 a^{4} + 9 a^{3} + a^{2} + 5 a + 8\right)\cdot 13 + \left(5 a^{4} + 8 a^{3} + 4 a^{2} + 6 a + 9\right)\cdot 13^{2} + \left(9 a^{4} + a^{3} + 9 a^{2} + 8 a + 8\right)\cdot 13^{3} + \left(2 a^{4} + 6 a^{3} + 8 a^{2}\right)\cdot 13^{4} + \left(3 a^{3} + 6 a^{2} + 8 a\right)\cdot 13^{5} + \left(11 a^{4} + 4 a^{2} + 5 a\right)\cdot 13^{6} + \left(5 a^{4} + 8 a^{3} + 12 a^{2} + 2 a + 3\right)\cdot 13^{7} + \left(6 a^{4} + 12 a^{3} + 5 a + 4\right)\cdot 13^{8} + \left(3 a^{4} + 10 a^{3} + 2 a^{2} + 10 a + 10\right)\cdot 13^{9} +O(13^{10})\) |
$r_{ 8 }$ | $=$ | \( 12 a^{3} + a^{2} + 8 a + \left(2 a^{4} + 11 a^{3} + 3 a^{2} + 11 a + 2\right)\cdot 13 + \left(4 a^{4} + 4 a^{3} + 12 a^{2} + 6 a + 9\right)\cdot 13^{2} + \left(10 a^{4} + 6 a^{3} + 11 a^{2} + a + 3\right)\cdot 13^{3} + \left(7 a^{4} + 11 a^{3} + 9 a^{2} + 7 a + 1\right)\cdot 13^{4} + \left(8 a^{4} + 10 a^{3} + 3 a + 6\right)\cdot 13^{5} + \left(a^{3} + 11 a^{2} + 8\right)\cdot 13^{6} + \left(7 a^{4} + 12 a^{2} + 2 a + 1\right)\cdot 13^{7} + \left(a^{4} + 11 a^{3} + 12 a^{2} + 6 a + 4\right)\cdot 13^{8} + \left(6 a^{4} + a^{3} + 7 a^{2} + 5 a + 8\right)\cdot 13^{9} +O(13^{10})\) |
$r_{ 9 }$ | $=$ | \( 8 a^{4} + 6 a^{3} + 7 a^{2} + 4 a + 8 + \left(9 a^{4} + 5 a^{3} + 6 a^{2} + 6\right)\cdot 13 + \left(11 a^{3} + 3 a^{2} + 10 a + 6\right)\cdot 13^{2} + \left(4 a^{4} + 9 a^{3} + 8 a^{2} + a + 8\right)\cdot 13^{3} + \left(8 a^{4} + 8 a^{3} + 7 a^{2} + 9 a + 8\right)\cdot 13^{4} + \left(3 a^{4} + 3 a^{3} + 9 a^{2} + 3 a + 9\right)\cdot 13^{5} + \left(10 a^{4} + 2 a^{3} + 12 a^{2} + 10 a + 5\right)\cdot 13^{6} + \left(6 a^{4} + 8 a^{3} + 6 a^{2} + 5 a + 11\right)\cdot 13^{7} + \left(3 a^{4} + 10 a^{3} + 12 a^{2} + 6 a + 6\right)\cdot 13^{8} + \left(3 a^{4} + 11 a^{3} + 12 a^{2} + 10 a + 3\right)\cdot 13^{9} +O(13^{10})\) |
$r_{ 10 }$ | $=$ | \( 8 a^{4} + 2 a^{3} + 8 a^{2} + 9 a + 10 + \left(7 a^{3} + 6 a^{2} + 11 a + 2\right)\cdot 13 + \left(5 a^{4} + a^{3} + 4 a^{2} + 4\right)\cdot 13^{2} + \left(2 a^{3} + a^{2} + 5 a + 8\right)\cdot 13^{3} + \left(9 a^{4} + 4 a^{3} + 7 a + 2\right)\cdot 13^{4} + \left(10 a^{4} + 12 a^{3} + 4 a^{2} + 11 a + 10\right)\cdot 13^{5} + \left(a^{4} + 6 a^{3} + 11 a^{2} + a + 1\right)\cdot 13^{6} + \left(a^{4} + 6 a^{3} + a^{2} + 7 a + 6\right)\cdot 13^{7} + \left(11 a^{4} + 6 a^{3} + 2 a^{2} + 2 a + 3\right)\cdot 13^{8} + \left(11 a^{4} + 9 a^{3} + 3 a^{2} + 3 a + 3\right)\cdot 13^{9} +O(13^{10})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,9)(2,7)(3,8)(4,6)(5,10)$ | $-2$ |
$5$ | $2$ | $(1,3)(2,6)(4,7)(5,10)(8,9)$ | $0$ |
$5$ | $2$ | $(1,8)(2,4)(3,9)(6,7)$ | $0$ |
$2$ | $5$ | $(1,6,7,8,10)(2,3,5,9,4)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ |
$2$ | $5$ | $(1,8,6,10,7)(2,9,3,4,5)$ | $\zeta_{5}^{3} + \zeta_{5}^{2}$ |
$2$ | $10$ | $(1,3,6,5,7,9,8,4,10,2)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
$2$ | $10$ | $(1,5,8,2,6,9,10,3,7,4)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.