Basic invariants
Dimension: | $9$ |
Group: | $M_{10}$ |
Conductor: | \(134217728000000\)\(\medspace = 2^{33} \cdot 5^{6} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 10.2.268435456000000.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 20T148 |
Parity: | even |
Determinant: | 1.8.2t1.a.a |
Projective image: | $A_6.C_2$ |
Projective stem field: | Galois closure of 10.2.268435456000000.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{10} - 4x^{9} - 2x^{8} + 16x^{7} + 4x^{6} + 16x^{5} - 88x^{4} - 96x^{3} + 196x^{2} + 176x - 232 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \( x^{5} + x + 14 \)
Roots:
$r_{ 1 }$ | $=$ | \( a^{4} + 3 a^{3} + 2 a^{2} + 13 a + 3 + \left(11 a^{4} + 14 a^{3} + 2 a^{2} + 7 a + 8\right)\cdot 17 + \left(2 a^{4} + 11 a^{3} + 2 a^{2} + 14 a + 4\right)\cdot 17^{2} + \left(13 a^{4} + 3 a^{3} + 9 a^{2} + 9 a + 13\right)\cdot 17^{3} + \left(8 a^{4} + 6 a^{3} + 6 a^{2} + 11 a + 1\right)\cdot 17^{4} + \left(7 a^{4} + 7 a^{3} + 5 a^{2} + 11 a + 13\right)\cdot 17^{5} + \left(8 a^{4} + 14 a^{3} + 15 a^{2} + 4 a + 7\right)\cdot 17^{6} + \left(13 a^{4} + 7 a^{3} + a^{2} + 4 a + 16\right)\cdot 17^{7} + \left(14 a^{4} + 10 a^{3} + a^{2} + 4 a + 4\right)\cdot 17^{8} + \left(2 a^{4} + 14 a^{3} + 2 a^{2} + 15 a + 4\right)\cdot 17^{9} +O(17^{10})\) |
$r_{ 2 }$ | $=$ | \( 3 a^{4} + 2 a^{3} + 4 a^{2} + 2 a + 8 + \left(14 a^{4} + 14 a^{2} + 3 a\right)\cdot 17 + \left(8 a^{4} + 8 a^{3} + 11 a^{2} + 9 a + 6\right)\cdot 17^{2} + \left(11 a^{4} + 11 a^{3} + 12 a^{2} + 9 a + 15\right)\cdot 17^{3} + \left(13 a^{4} + 13 a^{3} + 10 a^{2} + 14 a + 15\right)\cdot 17^{4} + \left(13 a^{4} + 15 a^{3} + a^{2} + 4 a + 7\right)\cdot 17^{5} + \left(2 a^{4} + 15 a^{3} + 6 a^{2} + 15 a + 13\right)\cdot 17^{6} + \left(13 a^{4} + 3 a^{3} + 11 a^{2} + 6 a + 12\right)\cdot 17^{7} + \left(2 a^{4} + 14 a^{2} + 9 a + 15\right)\cdot 17^{8} + \left(2 a^{4} + 8 a^{3} + 13 a^{2} + 8 a + 13\right)\cdot 17^{9} +O(17^{10})\) |
$r_{ 3 }$ | $=$ | \( 6 a^{4} + 4 a^{3} + a^{2} + 13 a + \left(7 a^{4} + 6 a^{3} + 14 a^{2} + 12 a + 10\right)\cdot 17 + \left(12 a^{4} + 13 a^{3} + 9 a^{2} + 13 a + 7\right)\cdot 17^{2} + \left(4 a^{4} + 9 a^{3} + 10 a^{2} + 4 a + 4\right)\cdot 17^{3} + \left(15 a^{4} + 7 a^{3} + 11 a^{2} + 15 a + 7\right)\cdot 17^{4} + \left(13 a^{4} + 15 a^{2} + 14 a + 7\right)\cdot 17^{5} + \left(5 a^{4} + a^{3} + 3 a^{2} + 15 a + 10\right)\cdot 17^{6} + \left(14 a^{4} + 16 a^{3} + 7 a^{2} + 9 a + 12\right)\cdot 17^{7} + \left(8 a^{4} + 10 a^{3} + 12 a^{2} + 16 a + 3\right)\cdot 17^{8} + \left(16 a^{4} + 8 a^{3} + 16 a^{2} + 11 a + 11\right)\cdot 17^{9} +O(17^{10})\) |
$r_{ 4 }$ | $=$ | \( 8 a^{4} + 11 a^{3} + 9 a^{2} + 13 a + 5 + \left(16 a^{4} + 7 a^{3} + 7 a^{2}\right)\cdot 17 + \left(a^{4} + 13 a^{3} + 8 a^{2} + 6 a + 6\right)\cdot 17^{2} + \left(14 a^{4} + 13 a^{3} + 6 a^{2} + 10 a + 15\right)\cdot 17^{3} + \left(13 a^{4} + 6 a^{3} + 3 a^{2} + 3 a + 12\right)\cdot 17^{4} + \left(a^{4} + 7 a^{3} + a^{2} + 7 a + 14\right)\cdot 17^{5} + \left(5 a^{4} + 10 a^{3} + 14 a^{2} + 4 a + 9\right)\cdot 17^{6} + \left(7 a^{4} + 8 a^{3} + 15 a^{2} + a + 3\right)\cdot 17^{7} + \left(10 a^{4} + 6 a^{3} + 5 a^{2} + 12 a + 15\right)\cdot 17^{8} + \left(15 a^{4} + 7 a^{3} + 3 a^{2} + 16 a + 13\right)\cdot 17^{9} +O(17^{10})\) |
$r_{ 5 }$ | $=$ | \( 9 a^{4} + 9 a^{3} + 11 a^{2} + 5 a + 6 + \left(6 a^{4} + 4 a^{3} + 15 a^{2} + 11\right)\cdot 17 + \left(3 a^{4} + 10 a^{3} + 8 a^{2} + 13 a + 1\right)\cdot 17^{2} + \left(7 a^{4} + 8 a^{3} + 10 a^{2} + 4 a + 5\right)\cdot 17^{3} + \left(3 a^{4} + 15 a^{3} + 14 a^{2} + 5 a + 4\right)\cdot 17^{4} + \left(5 a^{4} + 6 a^{3} + 14 a + 11\right)\cdot 17^{5} + \left(a^{4} + 4 a^{3} + 11 a^{2} + 12 a + 15\right)\cdot 17^{6} + \left(9 a^{4} + 6 a^{3} + 5 a^{2} + 5 a + 12\right)\cdot 17^{7} + \left(9 a^{4} + 8 a^{3} + 8 a^{2} + a\right)\cdot 17^{8} + \left(15 a^{4} + 9 a^{3} + 12 a^{2} + 12 a + 11\right)\cdot 17^{9} +O(17^{10})\) |
$r_{ 6 }$ | $=$ | \( 9 a^{4} + 10 a^{3} + 7 a^{2} + 5 a + 16 + \left(15 a^{4} + 9 a^{3} + 15 a^{2} + 15 a + 2\right)\cdot 17 + \left(14 a^{4} + 15 a^{3} + 16 a^{2} + 5 a + 6\right)\cdot 17^{2} + \left(10 a^{4} + 12 a^{3} + 2 a^{2} + 8 a + 9\right)\cdot 17^{3} + \left(4 a^{3} + 7 a^{2} + 15 a + 12\right)\cdot 17^{4} + \left(4 a^{4} + 15 a^{3} + a^{2} + 10 a + 9\right)\cdot 17^{5} + \left(4 a^{4} + 2 a^{3} + a^{2} + 7 a + 12\right)\cdot 17^{6} + \left(6 a^{4} + 13 a^{3} + 12 a^{2} + 6 a + 2\right)\cdot 17^{7} + \left(12 a^{3} + 9 a^{2} + 2 a + 7\right)\cdot 17^{8} + \left(6 a^{4} + 16 a^{3} + 10 a + 16\right)\cdot 17^{9} +O(17^{10})\) |
$r_{ 7 }$ | $=$ | \( 9 a^{4} + 13 a^{3} + 11 a^{2} + 6 a + 6 + \left(4 a^{4} + 5 a^{3} + 3 a^{2} + 4 a + 6\right)\cdot 17 + \left(4 a^{4} + 3 a^{3} + 4 a^{2} + 9 a + 2\right)\cdot 17^{2} + \left(9 a^{4} + 5 a^{3} + 8 a^{2} + 10\right)\cdot 17^{3} + \left(11 a^{4} + 13 a^{3} + 14 a^{2} + 10 a\right)\cdot 17^{4} + \left(15 a^{4} + 11 a^{3} + 9 a + 6\right)\cdot 17^{5} + \left(2 a^{4} + 3 a^{3} + 9 a^{2} + 9 a + 3\right)\cdot 17^{6} + \left(16 a^{4} + 8 a^{3} + 7 a^{2} + a + 15\right)\cdot 17^{7} + \left(16 a^{4} + 13 a^{3} + 7 a^{2} + 13 a + 16\right)\cdot 17^{8} + \left(8 a^{4} + 14 a^{3} + 11 a^{2} + 4 a + 15\right)\cdot 17^{9} +O(17^{10})\) |
$r_{ 8 }$ | $=$ | \( 12 a^{4} + 7 a^{3} + 6 a^{2} + 8 a + 5 + \left(14 a^{4} + 9 a^{3} + 15 a^{2} + a + 4\right)\cdot 17 + \left(14 a^{4} + 6 a^{2} + 5 a + 14\right)\cdot 17^{2} + \left(9 a^{4} + 5 a^{3} + 10 a^{2} + 9 a + 3\right)\cdot 17^{3} + \left(13 a^{4} + 2 a^{3} + 4 a^{2} + 9 a + 2\right)\cdot 17^{4} + \left(8 a^{4} + 9 a^{3} + 8 a^{2} + 10 a + 14\right)\cdot 17^{5} + \left(a^{4} + 12 a^{3} + 9 a^{2} + 8 a + 15\right)\cdot 17^{6} + \left(16 a^{4} + 7 a^{3} + 7 a^{2} + 15 a + 4\right)\cdot 17^{7} + \left(6 a^{4} + a^{3} + 2 a^{2} + 5 a + 12\right)\cdot 17^{8} + \left(4 a^{4} + 4 a^{3} + 11 a^{2} + 10 a + 15\right)\cdot 17^{9} +O(17^{10})\) |
$r_{ 9 }$ | $=$ | \( 13 a^{4} + a^{3} + 2 a + 9 + \left(4 a^{4} + 10 a^{3} + 8 a^{2} + 13 a + 4\right)\cdot 17 + \left(8 a^{4} + 8 a^{3} + 8 a^{2} + 8 a + 4\right)\cdot 17^{2} + \left(a^{4} + 13 a^{3} + 4 a^{2} + a + 5\right)\cdot 17^{3} + \left(5 a^{4} + 13 a^{3} + a^{2} + 14 a + 9\right)\cdot 17^{4} + \left(2 a^{4} + 13 a^{3} + 8 a^{2} + 4 a + 1\right)\cdot 17^{5} + \left(16 a^{4} + 9 a^{3} + 13 a^{2} + 3 a + 5\right)\cdot 17^{6} + \left(8 a^{4} + 6 a^{3} + 10 a^{2} + 13 a + 8\right)\cdot 17^{7} + \left(4 a^{4} + 5 a^{3} + 9 a^{2} + 11 a + 10\right)\cdot 17^{8} + \left(10 a^{4} + 16 a^{3} + 4 a^{2} + 15 a + 9\right)\cdot 17^{9} +O(17^{10})\) |
$r_{ 10 }$ | $=$ | \( 15 a^{4} + 8 a^{3} + a + 14 + \left(6 a^{4} + 6 a^{2} + 9 a + 2\right)\cdot 17 + \left(13 a^{4} + 7 a^{2} + 16 a + 15\right)\cdot 17^{2} + \left(2 a^{4} + a^{3} + 9 a^{2} + 8 a + 2\right)\cdot 17^{3} + \left(16 a^{4} + a^{3} + 10 a^{2} + 2 a + 1\right)\cdot 17^{4} + \left(11 a^{4} + 14 a^{3} + 7 a^{2} + 13 a + 16\right)\cdot 17^{5} + \left(2 a^{4} + 9 a^{3} + a^{2} + 2 a + 7\right)\cdot 17^{6} + \left(14 a^{4} + 6 a^{3} + 5 a^{2} + 3 a + 12\right)\cdot 17^{7} + \left(9 a^{4} + 15 a^{3} + 13 a^{2} + 8 a + 14\right)\cdot 17^{8} + \left(2 a^{4} + a^{3} + 8 a^{2} + 13 a + 6\right)\cdot 17^{9} +O(17^{10})\) |
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 |
$1$ | $1$ | $()$ | $9$ |
$45$ | $2$ | $(1,5)(2,3)(4,7)(8,9)$ | $1$ |
$80$ | $3$ | $(1,3,9)(2,5,8)(4,7,6)$ | $0$ |
$90$ | $4$ | $(1,3,5,2)(4,9,7,8)$ | $1$ |
$180$ | $4$ | $(1,4,5,7)(2,9,3,8)$ | $-1$ |
$144$ | $5$ | $(1,8,7,5,2)(3,9,10,6,4)$ | $-1$ |
$90$ | $8$ | $(1,5,9,8,4,10,6,2)(3,7)$ | $1$ |
$90$ | $8$ | $(1,10,9,2,4,5,6,8)(3,7)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.