Basic invariants
Dimension: | $7$ |
Group: | $C_2^3:(C_7: C_3)$ |
Conductor: | \(525346636864\)\(\medspace = 2^{6} \cdot 7^{4} \cdot 43^{4} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 8.0.525346636864.4 |
Galois orbit size: | $1$ |
Smallest permutation container: | $C_2^3:(C_7: C_3)$ |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $F_8:C_3$ |
Projective stem field: | Galois closure of 8.0.525346636864.4 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 2x^{7} + 2x^{6} - 12x^{5} + 64x^{4} - 184x^{3} + 296x^{2} - 280x + 128 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \( x^{3} + 2x + 11 \)
Roots:
$r_{ 1 }$ | $=$ | \( 1 + 12\cdot 13 + 11\cdot 13^{2} + 2\cdot 13^{3} + 10\cdot 13^{4} + 2\cdot 13^{5} + 6\cdot 13^{6} + 6\cdot 13^{7} + 10\cdot 13^{8} + 2\cdot 13^{9} +O(13^{10})\) |
$r_{ 2 }$ | $=$ | \( 4 + 8\cdot 13 + 9\cdot 13^{2} + 11\cdot 13^{3} + 9\cdot 13^{4} + 8\cdot 13^{5} + 9\cdot 13^{6} + 8\cdot 13^{7} + 11\cdot 13^{8} + 13^{9} +O(13^{10})\) |
$r_{ 3 }$ | $=$ | \( 12 a^{2} + 7 a + 10 + \left(3 a^{2} + 6 a + 6\right)\cdot 13 + \left(4 a^{2} + 9 a + 1\right)\cdot 13^{2} + \left(6 a^{2} + a + 7\right)\cdot 13^{3} + \left(5 a^{2} + 4 a + 9\right)\cdot 13^{4} + \left(10 a^{2} + 12\right)\cdot 13^{5} + \left(10 a^{2} + 4 a\right)\cdot 13^{6} + \left(10 a^{2} + 12 a + 2\right)\cdot 13^{7} + \left(6 a^{2} + 2 a + 1\right)\cdot 13^{8} + \left(11 a^{2} + 4 a + 9\right)\cdot 13^{9} +O(13^{10})\) |
$r_{ 4 }$ | $=$ | \( 4 a^{2} + 12 a + 6 + \left(9 a^{2} + 2 a + 8\right)\cdot 13 + \left(8 a^{2} + 3 a + 8\right)\cdot 13^{2} + \left(10 a^{2} + 10\right)\cdot 13^{3} + \left(11 a^{2} + a + 6\right)\cdot 13^{4} + \left(10 a^{2} + 8 a + 7\right)\cdot 13^{5} + \left(a^{2} + 4 a + 10\right)\cdot 13^{6} + \left(a^{2} + a + 8\right)\cdot 13^{7} + \left(11 a^{2} + 9 a + 6\right)\cdot 13^{8} + \left(3 a^{2} + 6 a + 5\right)\cdot 13^{9} +O(13^{10})\) |
$r_{ 5 }$ | $=$ | \( 2 a^{2} + 2 a + 12 + \left(5 a^{2} + 11 a + 2\right)\cdot 13 + \left(11 a^{2} + 6 a + 12\right)\cdot 13^{2} + \left(11 a^{2} + 3 a + 7\right)\cdot 13^{3} + \left(6 a^{2} + 6 a + 4\right)\cdot 13^{4} + 2\cdot 13^{5} + \left(11 a^{2} + 4 a + 1\right)\cdot 13^{6} + \left(11 a^{2} + 2 a + 10\right)\cdot 13^{7} + \left(11 a^{2} + a + 7\right)\cdot 13^{8} + \left(10 a^{2} + a + 10\right)\cdot 13^{9} +O(13^{10})\) |
$r_{ 6 }$ | $=$ | \( 10 a^{2} + 8 a + 3 + \left(6 a^{2} + 6\right)\cdot 13 + \left(12 a^{2} + 3 a + 12\right)\cdot 13^{2} + \left(8 a^{2} + 4 a + 1\right)\cdot 13^{3} + \left(11 a^{2} + 7 a + 9\right)\cdot 13^{4} + \left(7 a^{2} + 7 a\right)\cdot 13^{5} + \left(9 a^{2} + 8\right)\cdot 13^{6} + \left(8 a^{2} + 3 a + 3\right)\cdot 13^{7} + \left(2 a^{2} + 7 a + 4\right)\cdot 13^{8} + \left(2 a^{2} + a + 5\right)\cdot 13^{9} +O(13^{10})\) |
$r_{ 7 }$ | $=$ | \( 7 a^{2} + 12 a + 10 + \left(11 a^{2} + 11 a + 2\right)\cdot 13 + \left(5 a^{2} + 2 a + 9\right)\cdot 13^{2} + \left(3 a^{2} + 9 a + 9\right)\cdot 13^{3} + \left(7 a^{2} + 5 a\right)\cdot 13^{4} + \left(a^{2} + 4 a + 8\right)\cdot 13^{5} + \left(4 a + 12\right)\cdot 13^{6} + \left(9 a + 2\right)\cdot 13^{7} + \left(3 a^{2} + 2 a\right)\cdot 13^{8} + \left(11 a^{2} + 5 a + 11\right)\cdot 13^{9} +O(13^{10})\) |
$r_{ 8 }$ | $=$ | \( 4 a^{2} + 11 a + 8 + \left(2 a^{2} + 5 a + 4\right)\cdot 13 + \left(9 a^{2} + 12\right)\cdot 13^{2} + \left(10 a^{2} + 7 a + 12\right)\cdot 13^{3} + \left(8 a^{2} + a\right)\cdot 13^{4} + \left(7 a^{2} + 5 a + 9\right)\cdot 13^{5} + \left(5 a^{2} + 8 a + 2\right)\cdot 13^{6} + \left(6 a^{2} + 10 a + 9\right)\cdot 13^{7} + \left(3 a^{2} + 2 a + 9\right)\cdot 13^{8} + \left(12 a^{2} + 7 a + 5\right)\cdot 13^{9} +O(13^{10})\) |
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 |
$1$ | $1$ | $()$ | $7$ |
$7$ | $2$ | $(1,8)(2,7)(3,4)(5,6)$ | $-1$ |
$28$ | $3$ | $(1,5,4)(2,3,6)$ | $1$ |
$28$ | $3$ | $(1,4,5)(2,6,3)$ | $1$ |
$28$ | $6$ | $(1,6,5,2,4,3)(7,8)$ | $-1$ |
$28$ | $6$ | $(1,3,4,2,5,6)(7,8)$ | $-1$ |
$24$ | $7$ | $(1,3,4,6,2,8,5)$ | $0$ |
$24$ | $7$ | $(1,6,5,4,8,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.