Basic invariants
Dimension: | $7$ |
Group: | $C_2^3:C_7$ |
Conductor: | \(594823321\)\(\medspace = 29^{6} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 8.0.594823321.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $C_2^3:C_7$ |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $F_8$ |
Projective stem field: | Galois closure of 8.0.594823321.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 3x^{7} + 9x^{6} - 15x^{5} + 23x^{4} - 21x^{3} + 18x^{2} - 7x + 2 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \( x^{7} + 4x + 9 \)
Roots:
$r_{ 1 }$ | $=$ | \( 10 + 4\cdot 11 + 10\cdot 11^{2} + 10\cdot 11^{3} + 8\cdot 11^{4} + 3\cdot 11^{5} + 6\cdot 11^{6} + 11^{7} + 10\cdot 11^{8} + 4\cdot 11^{9} +O(11^{10})\) |
$r_{ 2 }$ | $=$ | \( a^{6} + 10 a^{5} + 6 a^{4} + 4 a^{3} + 6 a^{2} + 2 a + 4 + \left(9 a^{6} + 8 a^{5} + 8 a^{4} + 5 a^{3} + 3 a^{2} + a + 5\right)\cdot 11 + \left(8 a^{6} + a^{5} + 8 a^{3} + 5 a^{2} + 6 a\right)\cdot 11^{2} + \left(4 a^{6} + 3 a^{5} + 8 a^{3} + 10 a^{2} + 10 a + 7\right)\cdot 11^{3} + \left(3 a^{6} + 8 a^{5} + 8 a^{4} + 8 a^{2} + a + 2\right)\cdot 11^{4} + \left(3 a^{6} + 6 a^{5} + 5 a^{4} + 4 a^{3} + 4 a^{2} + 2 a + 9\right)\cdot 11^{5} + \left(8 a^{6} + 3 a^{5} + 7 a^{4} + 6 a^{3} + 6 a^{2} + 9 a\right)\cdot 11^{6} + \left(3 a^{6} + 10 a^{5} + a^{4} + 3 a^{3} + 5 a^{2} + 8 a\right)\cdot 11^{7} + \left(5 a^{6} + 2 a^{5} + 2 a^{4} + 8 a^{3} + 2 a^{2} + 6 a + 9\right)\cdot 11^{8} + \left(7 a^{6} + 4 a^{5} + 10 a^{4} + 4 a^{3} + 5 a + 10\right)\cdot 11^{9} +O(11^{10})\) |
$r_{ 3 }$ | $=$ | \( 2 a^{6} + 9 a^{5} + 6 a^{4} + 7 a^{3} + 2 a^{2} + 7 a + 9 + \left(4 a^{6} + 7 a^{5} + 5 a^{4} + a^{3} + 2 a^{2} + 7 a + 5\right)\cdot 11 + \left(10 a^{6} + 10 a^{5} + 3 a^{4} + a^{3} + 2 a + 10\right)\cdot 11^{2} + \left(5 a^{6} + 4 a^{5} + 7 a^{4} + 6 a^{3} + a^{2} + 4 a + 10\right)\cdot 11^{3} + \left(2 a^{5} + 2 a^{4} + 7 a^{3} + 6 a^{2} + 2 a + 9\right)\cdot 11^{4} + \left(8 a^{5} + 9 a^{4} + a^{3} + 6 a^{2} + 4 a + 5\right)\cdot 11^{5} + \left(5 a^{6} + 2 a^{5} + 3 a^{4} + 8 a^{3} + 9 a^{2} + 10 a\right)\cdot 11^{6} + \left(8 a^{6} + a^{5} + 9 a^{4} + 8 a^{3} + a^{2} + 2 a + 2\right)\cdot 11^{7} + \left(7 a^{6} + 10 a^{5} + 7 a^{4} + 2 a^{3} + 9 a^{2} + 3 a + 3\right)\cdot 11^{8} + \left(4 a^{6} + a^{5} + 5 a^{4} + 9 a^{3} + 7 a^{2} + 3 a + 1\right)\cdot 11^{9} +O(11^{10})\) |
$r_{ 4 }$ | $=$ | \( 3 a^{6} + 10 a^{5} + 5 a^{4} + 3 a^{3} + 4 a^{2} + 7 a + 3 + \left(6 a^{6} + 9 a^{5} + a^{4} + 5 a^{3} + 3 a^{2} + 2 a + 8\right)\cdot 11 + \left(a^{6} + 9 a^{5} + a^{4} + 9 a^{3} + 7 a^{2} + 4 a\right)\cdot 11^{2} + \left(9 a^{6} + 6 a^{5} + 9 a^{4} + 5 a^{3} + 9 a^{2} + 5 a + 3\right)\cdot 11^{3} + \left(9 a^{6} + a^{4} + 9 a^{3} + 2 a^{2} + 8 a + 7\right)\cdot 11^{4} + \left(5 a^{6} + 2 a^{4} + 3 a^{3} + 8 a + 8\right)\cdot 11^{5} + \left(10 a^{6} + 10 a^{5} + 7 a^{4} + 9 a^{3} + 10 a^{2} + 3\right)\cdot 11^{6} + \left(4 a^{6} + 6 a^{5} + 3 a^{4} + 8 a^{3} + a^{2} + 9 a + 7\right)\cdot 11^{7} + \left(3 a^{6} + 2 a^{5} + 9 a^{4} + a^{3} + 9 a + 5\right)\cdot 11^{8} + \left(5 a^{6} + 5 a^{5} + 6 a^{4} + 10 a^{3} + 5 a^{2} + 6\right)\cdot 11^{9} +O(11^{10})\) |
$r_{ 5 }$ | $=$ | \( 9 a^{6} + 2 a^{4} + 9 a^{3} + 10 a^{2} + 7 a + \left(10 a^{6} + 8 a^{4} + 10 a^{3} + 7 a^{2} + a + 5\right)\cdot 11 + \left(6 a^{6} + 10 a^{4} + a^{3} + a^{2} + 6 a\right)\cdot 11^{2} + \left(6 a^{6} + 3 a^{5} + 9 a^{4} + 10 a^{3} + 9 a^{2} + 2 a + 7\right)\cdot 11^{3} + \left(5 a^{6} + 9 a^{5} + 3 a^{4} + 4 a^{3} + 2 a\right)\cdot 11^{4} + \left(3 a^{6} + 8 a^{5} + 7 a^{4} + 3 a^{3} + 8 a^{2} + 8 a + 2\right)\cdot 11^{5} + \left(2 a^{6} + 7 a^{5} + 9 a^{4} + 3 a^{3} + 7 a^{2} + 8 a + 10\right)\cdot 11^{6} + \left(5 a^{6} + 2 a^{5} + 7 a^{4} + 2 a^{2} + 3 a + 1\right)\cdot 11^{7} + \left(5 a^{6} + 7 a^{5} + 9 a^{4} + 6 a^{3} + 8 a^{2} + 4 a + 3\right)\cdot 11^{8} + \left(a^{6} + 9 a^{5} + 4 a^{4} + 10 a^{2} + 1\right)\cdot 11^{9} +O(11^{10})\) |
$r_{ 6 }$ | $=$ | \( 9 a^{6} + 3 a^{5} + 2 a^{4} + 7 a^{3} + 9 a^{2} + 4 a + \left(2 a^{6} + 6 a^{5} + 10 a^{4} + 3 a^{3} + 2 a^{2} + 2 a + 9\right)\cdot 11 + \left(4 a^{6} + 4 a^{5} + 10 a^{4} + 4 a^{3} + 10 a + 6\right)\cdot 11^{2} + \left(8 a^{6} + 3 a^{5} + 2 a^{4} + 4 a^{3} + 4 a^{2} + 3\right)\cdot 11^{3} + \left(2 a^{6} + 9 a^{5} + 3 a^{4} + 3 a^{3} + 7 a^{2} + 5 a\right)\cdot 11^{4} + \left(10 a^{6} + 10 a^{5} + 6 a^{3} + 7 a^{2} + 5 a\right)\cdot 11^{5} + \left(a^{6} + 6 a^{5} + 3 a^{4} + 6 a^{3} + a^{2} + 1\right)\cdot 11^{6} + \left(6 a^{6} + 6 a^{5} + 3 a^{4} + a^{3} + 4 a^{2} + 6 a + 2\right)\cdot 11^{7} + \left(a^{6} + 6 a^{5} + 7 a^{4} + 3 a^{3} + 3 a^{2} + 10\right)\cdot 11^{8} + \left(4 a^{6} + 4 a^{5} + 2 a^{4} + 2 a^{3} + 8 a^{2} + 10 a\right)\cdot 11^{9} +O(11^{10})\) |
$r_{ 7 }$ | $=$ | \( 10 a^{6} + 6 a^{5} + 4 a^{4} + 5 a^{3} + a + 5 + \left(5 a^{6} + a^{5} + 5 a^{4} + 9 a^{3} + 8 a^{2} + 5 a + 5\right)\cdot 11 + \left(4 a^{6} + 6 a^{5} + 8 a^{4} + 5 a^{3} + 10 a^{2} + 3 a + 1\right)\cdot 11^{2} + \left(a^{6} + 2 a^{5} + 10 a^{4} + 5 a^{3} + 3 a^{2} + 8 a\right)\cdot 11^{3} + \left(7 a^{6} + 2 a^{5} + 5 a^{3} + 10 a^{2} + 2 a + 9\right)\cdot 11^{4} + \left(4 a^{6} + 5 a^{5} + 3 a^{4} + 4 a^{3} + 5 a^{2} + 5 a + 2\right)\cdot 11^{5} + \left(2 a^{6} + 8 a^{5} + a^{4} + 7 a^{3} + a^{2} + 5 a + 4\right)\cdot 11^{6} + \left(9 a^{6} + 6 a^{5} + 10 a^{4} + 5 a^{3} + 6 a^{2} + a + 9\right)\cdot 11^{7} + \left(6 a^{6} + 5 a^{5} + 9 a^{4} + 7 a^{3} + 7 a^{2} + 4 a + 7\right)\cdot 11^{8} + \left(5 a^{6} + a^{5} + 5 a^{4} + 6 a^{3} + 7 a^{2} + 7 a + 10\right)\cdot 11^{9} +O(11^{10})\) |
$r_{ 8 }$ | $=$ | \( 10 a^{6} + 6 a^{5} + 8 a^{4} + 9 a^{3} + 2 a^{2} + 5 a + 5 + \left(4 a^{6} + 9 a^{5} + 4 a^{4} + 7 a^{3} + 5 a^{2} + a\right)\cdot 11 + \left(7 a^{6} + 10 a^{5} + 8 a^{4} + a^{3} + 7 a^{2} + 2\right)\cdot 11^{2} + \left(7 a^{6} + 8 a^{5} + 3 a^{4} + 3 a^{3} + 5 a^{2} + a + 1\right)\cdot 11^{3} + \left(3 a^{6} + a^{4} + a^{3} + 7 a^{2} + 10 a + 5\right)\cdot 11^{4} + \left(5 a^{6} + 4 a^{5} + 5 a^{4} + 9 a^{3} + 10 a^{2} + 9 a\right)\cdot 11^{5} + \left(2 a^{6} + 4 a^{5} + 2 a^{3} + 6 a^{2} + 8 a + 6\right)\cdot 11^{6} + \left(6 a^{6} + 9 a^{5} + 8 a^{4} + 4 a^{3} + 10 a^{2} + 8\right)\cdot 11^{7} + \left(2 a^{6} + 8 a^{5} + 8 a^{4} + 3 a^{3} + a^{2} + 4 a + 5\right)\cdot 11^{8} + \left(4 a^{6} + 5 a^{5} + 7 a^{4} + 10 a^{3} + 4 a^{2} + 5 a + 7\right)\cdot 11^{9} +O(11^{10})\) |
Generators of the action on the roots $r_1, \ldots, r_{ 8 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 8 }$ | Character value |
$1$ | $1$ | $()$ | $7$ |
$7$ | $2$ | $(1,8)(2,7)(3,4)(5,6)$ | $-1$ |
$8$ | $7$ | $(1,7,8,3,4,2,6)$ | $0$ |
$8$ | $7$ | $(1,8,4,6,7,3,2)$ | $0$ |
$8$ | $7$ | $(1,3,6,8,2,7,4)$ | $0$ |
$8$ | $7$ | $(1,4,7,2,8,6,3)$ | $0$ |
$8$ | $7$ | $(1,2,3,7,6,4,8)$ | $0$ |
$8$ | $7$ | $(1,6,2,4,3,8,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.