Basic invariants
Dimension: | $7$ |
Group: | $C_2^3:(C_7: C_3)$ |
Conductor: | \(243336191536233\)\(\medspace = 3^{4} \cdot 313^{5} \) |
Artin stem field: | Galois closure of 8.0.777431921841.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | 24T283 |
Parity: | even |
Determinant: | 1.313.3t1.a.a |
Projective image: | $F_8:C_3$ |
Projective stem field: | Galois closure of 8.0.777431921841.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 9x^{6} - 4x^{5} + 24x^{4} + 21x^{3} + 22x^{2} - 3x + 3 \) . |
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:
$r_{ 1 }$ | $=$ |
\( 1 + 3\cdot 11 + 3\cdot 11^{2} + 8\cdot 11^{3} + 5\cdot 11^{4} + 3\cdot 11^{5} + 11^{6} + 3\cdot 11^{7} + 3\cdot 11^{8} + 11^{9} +O(11^{10})\)
$r_{ 2 }$ |
$=$ |
\( 8 a^{2} + 6 a + 3 + \left(8 a^{2} + 7 a + 7\right)\cdot 11 + \left(2 a^{2} + 7\right)\cdot 11^{2} + \left(5 a + 4\right)\cdot 11^{3} + \left(9 a^{2} + 2 a + 1\right)\cdot 11^{4} + \left(2 a^{2} + 10 a + 5\right)\cdot 11^{5} + \left(2 a^{2} + 4 a + 10\right)\cdot 11^{6} + \left(9 a^{2} + 5 a + 1\right)\cdot 11^{7} + \left(9 a^{2} + 2 a + 9\right)\cdot 11^{8} + \left(10 a^{2} + 6 a + 6\right)\cdot 11^{9} +O(11^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 2 a^{2} + a + 6 + \left(2 a^{2} + a + 9\right)\cdot 11 + \left(8 a^{2} + 4 a + 3\right)\cdot 11^{2} + \left(2 a^{2} + 10 a + 4\right)\cdot 11^{3} + \left(4 a^{2} + 6 a + 2\right)\cdot 11^{4} + \left(4 a^{2} + 6 a + 7\right)\cdot 11^{5} + \left(7 a^{2} + 6 a + 2\right)\cdot 11^{6} + \left(9 a^{2} + 4 a + 6\right)\cdot 11^{7} + \left(a^{2} + 6 a + 9\right)\cdot 11^{8} + \left(a^{2} + 2 a + 4\right)\cdot 11^{9} +O(11^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 9 + 10\cdot 11 + 3\cdot 11^{2} + 6\cdot 11^{3} + 8\cdot 11^{5} + 11^{6} + 8\cdot 11^{7} + 10\cdot 11^{8} + 8\cdot 11^{9} +O(11^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 10 a^{2} + 8 a + 3 + \left(a + 8\right)\cdot 11 + \left(9 a^{2} + 5\right)\cdot 11^{2} + \left(5 a^{2} + 2 a + 9\right)\cdot 11^{3} + \left(6 a^{2} + 3 a + 9\right)\cdot 11^{4} + 2\cdot 11^{5} + \left(6 a^{2} + 8 a + 3\right)\cdot 11^{6} + \left(9 a^{2} + 9 a + 8\right)\cdot 11^{7} + \left(a^{2} + 7 a + 1\right)\cdot 11^{8} + \left(3 a^{2} + 10 a + 1\right)\cdot 11^{9} +O(11^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 10 a^{2} + 3 + \left(2 a^{2} + a + 7\right)\cdot 11 + \left(8 a^{2} + 3 a + 4\right)\cdot 11^{2} + \left(4 a^{2} + 7 a + 4\right)\cdot 11^{3} + \left(8 a^{2} + 1\right)\cdot 11^{4} + \left(3 a^{2} + 8 a + 7\right)\cdot 11^{5} + \left(6 a^{2} + 6 a + 3\right)\cdot 11^{6} + \left(a^{2} + 8 a + 1\right)\cdot 11^{7} + \left(4 a^{2} + 5 a + 1\right)\cdot 11^{8} + \left(4 a^{2} + 9 a + 10\right)\cdot 11^{9} +O(11^{10})\)
| $r_{ 7 }$ |
$=$ |
\( 2 a^{2} + 3 a + 7 + \left(7 a^{2} + 8 a + 5\right)\cdot 11 + \left(4 a^{2} + 7 a + 3\right)\cdot 11^{2} + \left(a + 2\right)\cdot 11^{3} + \left(7 a^{2} + 7 a + 3\right)\cdot 11^{4} + \left(6 a^{2} + 2 a + 7\right)\cdot 11^{5} + \left(9 a^{2} + 7 a\right)\cdot 11^{6} + \left(10 a^{2} + 3 a + 10\right)\cdot 11^{7} + \left(4 a^{2} + 8 a + 5\right)\cdot 11^{8} + \left(3 a^{2} + a + 1\right)\cdot 11^{9} +O(11^{10})\)
| $r_{ 8 }$ |
$=$ |
\( a^{2} + 4 a + 1 + \left(2 a + 3\right)\cdot 11 + 6 a\cdot 11^{2} + \left(8 a^{2} + 6 a + 4\right)\cdot 11^{3} + \left(8 a^{2} + a + 8\right)\cdot 11^{4} + \left(3 a^{2} + 5 a + 2\right)\cdot 11^{5} + \left(a^{2} + 10 a + 9\right)\cdot 11^{6} + \left(3 a^{2} + 4\right)\cdot 11^{7} + \left(10 a^{2} + 2 a + 2\right)\cdot 11^{8} + \left(9 a^{2} + 2 a + 9\right)\cdot 11^{9} +O(11^{10})\)
| |
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 |
$1$ | $1$ | $()$ | $7$ |
$7$ | $2$ | $(1,4)(2,5)(3,7)(6,8)$ | $-1$ |
$28$ | $3$ | $(2,4,5)(3,7,6)$ | $\zeta_{3}$ |
$28$ | $3$ | $(2,5,4)(3,6,7)$ | $-\zeta_{3} - 1$ |
$28$ | $6$ | $(1,8,4,2,7,5)(3,6)$ | $-\zeta_{3}$ |
$28$ | $6$ | $(1,5,7,2,4,8)(3,6)$ | $\zeta_{3} + 1$ |
$24$ | $7$ | $(1,2,5,6,7,4,8)$ | $0$ |
$24$ | $7$ | $(1,6,8,5,4,2,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.