Basic invariants
Dimension: | $7$ |
Group: | $C_2^3:(C_7: C_3)$ |
Conductor: | \(8491301244928\)\(\medspace = 2^{12} \cdot 73^{5} \) |
Artin stem field: | Galois closure of 8.0.116319195136.3 |
Galois orbit size: | $2$ |
Smallest permutation container: | 24T283 |
Parity: | even |
Determinant: | 1.73.3t1.a.a |
Projective image: | $F_8:C_3$ |
Projective stem field: | Galois closure of 8.0.116319195136.3 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 2x^{7} + 10x^{6} - 6x^{5} + 2x^{4} + 8x^{3} - 16x^{2} + 16x + 26 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \( x^{3} + 2x + 9 \)
Roots:
$r_{ 1 }$ | $=$ | \( a^{2} + 9 a + 10 + \left(7 a + 8\right)\cdot 11 + \left(8 a^{2} + 4\right)\cdot 11^{2} + \left(6 a^{2} + 7 a + 4\right)\cdot 11^{3} + \left(5 a^{2} + 6 a + 2\right)\cdot 11^{4} + \left(7 a^{2} + 9 a + 9\right)\cdot 11^{5} + \left(9 a^{2} + a + 10\right)\cdot 11^{6} + \left(9 a^{2} + 2 a\right)\cdot 11^{7} + \left(9 a^{2} + 2 a + 3\right)\cdot 11^{8} + \left(9 a^{2} + 5 a + 10\right)\cdot 11^{9} +O(11^{10})\) |
$r_{ 2 }$ | $=$ | \( 4 + 4\cdot 11 + 7\cdot 11^{2} + 4\cdot 11^{4} + 6\cdot 11^{5} + 9\cdot 11^{6} + 9\cdot 11^{7} + 11^{8} +O(11^{10})\) |
$r_{ 3 }$ | $=$ | \( 8 a + 8 + \left(a^{2} + 6\right)\cdot 11 + \left(5 a^{2} + 7 a + 3\right)\cdot 11^{2} + \left(6 a^{2} + 5 a + 6\right)\cdot 11^{3} + \left(a^{2} + 4 a + 3\right)\cdot 11^{4} + \left(5 a^{2} + 9 a + 4\right)\cdot 11^{5} + \left(2 a^{2} + 6 a + 1\right)\cdot 11^{6} + \left(a^{2} + 8 a + 10\right)\cdot 11^{7} + \left(5 a^{2} + 5 a + 1\right)\cdot 11^{8} + \left(10 a^{2} + 6 a + 10\right)\cdot 11^{9} +O(11^{10})\) |
$r_{ 4 }$ | $=$ | \( 3 a^{2} + 7 a + 9 + \left(10 a^{2} + 2 a + 7\right)\cdot 11 + \left(2 a^{2} + 6 a + 1\right)\cdot 11^{2} + \left(6 a^{2} + 4 a\right)\cdot 11^{3} + \left(7 a^{2} + 4 a + 5\right)\cdot 11^{4} + \left(2 a^{2} + a + 6\right)\cdot 11^{5} + \left(3 a^{2} + 9\right)\cdot 11^{6} + \left(a^{2} + 6 a + 7\right)\cdot 11^{7} + \left(6 a^{2} + 1\right)\cdot 11^{8} + \left(4 a^{2} + a + 3\right)\cdot 11^{9} +O(11^{10})\) |
$r_{ 5 }$ | $=$ | \( 3 a^{2} + 6 a + 1 + \left(9 a^{2} + 4 a + 3\right)\cdot 11 + \left(3 a^{2} + 4 a + 9\right)\cdot 11^{2} + \left(6 a^{2} + 2 a + 9\right)\cdot 11^{3} + \left(3 a^{2} + 8 a + 9\right)\cdot 11^{4} + \left(5 a^{2} + a\right)\cdot 11^{5} + \left(4 a^{2} + 3 a + 4\right)\cdot 11^{6} + \left(9 a^{2} + 5 a + 6\right)\cdot 11^{7} + \left(10 a^{2} + 4 a + 9\right)\cdot 11^{8} + \left(2 a^{2} + 8 a + 3\right)\cdot 11^{9} +O(11^{10})\) |
$r_{ 6 }$ | $=$ | \( 7 a^{2} + 6 a + 7 + 9\cdot 11 + \left(4 a + 8\right)\cdot 11^{2} + \left(9 a^{2} + 10 a + 3\right)\cdot 11^{3} + \left(8 a^{2} + 10 a + 10\right)\cdot 11^{4} + \left(10 a + 3\right)\cdot 11^{5} + \left(9 a^{2} + 8 a + 6\right)\cdot 11^{6} + \left(10 a^{2} + 2 a + 9\right)\cdot 11^{7} + \left(5 a^{2} + 8 a + 8\right)\cdot 11^{8} + \left(7 a^{2} + 4 a + 10\right)\cdot 11^{9} +O(11^{10})\) |
$r_{ 7 }$ | $=$ | \( 8 a^{2} + 8 a + 4 + \left(5 a + 6\right)\cdot 11 + \left(2 a^{2} + 10 a + 10\right)\cdot 11^{2} + \left(9 a^{2} + 2 a + 9\right)\cdot 11^{3} + \left(5 a^{2} + 9 a + 1\right)\cdot 11^{4} + \left(10 a + 9\right)\cdot 11^{5} + \left(4 a^{2} + 10\right)\cdot 11^{6} + \left(8 a + 8\right)\cdot 11^{7} + \left(6 a^{2} + 6\right)\cdot 11^{8} + \left(8 a^{2} + 7 a + 7\right)\cdot 11^{9} +O(11^{10})\) |
$r_{ 8 }$ | $=$ | \( 3 + 8\cdot 11 + 8\cdot 11^{2} + 8\cdot 11^{3} + 6\cdot 11^{4} + 3\cdot 11^{5} + 2\cdot 11^{6} + 11^{7} + 10\cdot 11^{8} + 8\cdot 11^{9} +O(11^{10})\) |
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 |
$1$ | $1$ | $()$ | $7$ |
$7$ | $2$ | $(1,3)(2,8)(4,5)(6,7)$ | $-1$ |
$28$ | $3$ | $(1,4,2)(3,5,8)$ | $\zeta_{3}$ |
$28$ | $3$ | $(1,2,4)(3,8,5)$ | $-\zeta_{3} - 1$ |
$28$ | $6$ | $(1,5,2,3,4,8)(6,7)$ | $-\zeta_{3}$ |
$28$ | $6$ | $(1,8,4,3,2,5)(6,7)$ | $\zeta_{3} + 1$ |
$24$ | $7$ | $(2,5,6,4,3,7,8)$ | $0$ |
$24$ | $7$ | $(2,4,8,6,7,5,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.