Basic invariants
Dimension: | $2$ |
Group: | $D_{7}$ |
Conductor: | \(2008\)\(\medspace = 2^{3} \cdot 251 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.1.8096384512.1 |
Galois orbit size: | $3$ |
Smallest permutation container: | $D_{7}$ |
Parity: | odd |
Determinant: | 1.2008.2t1.b.a |
Projective image: | $D_7$ |
Projective stem field: | Galois closure of 7.1.8096384512.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - 3x^{6} + 6x^{5} - 14x^{4} + 15x^{3} - x^{2} + 82x - 134 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: \( x^{2} + 45x + 5 \)
Roots:
$r_{ 1 }$ | $=$ | \( 17 a + 15 + \left(41 a + 5\right)\cdot 47 + \left(16 a + 23\right)\cdot 47^{2} + \left(22 a + 6\right)\cdot 47^{3} + \left(41 a + 46\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 2 }$ | $=$ | \( 46 a + 22 + \left(12 a + 26\right)\cdot 47 + \left(18 a + 26\right)\cdot 47^{2} + \left(29 a + 44\right)\cdot 47^{3} + \left(38 a + 31\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 3 }$ | $=$ | \( 30 a + 2 + \left(5 a + 24\right)\cdot 47 + \left(30 a + 15\right)\cdot 47^{2} + \left(24 a + 34\right)\cdot 47^{3} + \left(5 a + 12\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 4 }$ | $=$ | \( a + 20 + \left(34 a + 6\right)\cdot 47 + \left(28 a + 3\right)\cdot 47^{2} + \left(17 a + 38\right)\cdot 47^{3} + \left(8 a + 32\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 5 }$ | $=$ | \( 2 a + 22 + \left(36 a + 26\right)\cdot 47 + \left(23 a + 15\right)\cdot 47^{2} + \left(4 a + 10\right)\cdot 47^{3} + \left(6 a + 31\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 6 }$ | $=$ | \( 37 + 2\cdot 47 + 30\cdot 47^{2} + 11\cdot 47^{3} + 41\cdot 47^{4} +O(47^{5})\) |
$r_{ 7 }$ | $=$ | \( 45 a + 26 + \left(10 a + 2\right)\cdot 47 + \left(23 a + 27\right)\cdot 47^{2} + \left(42 a + 42\right)\cdot 47^{3} + \left(40 a + 38\right)\cdot 47^{4} +O(47^{5})\) |
Generators of the action on the roots $r_1, \ldots, r_{ 7 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 7 }$ | Character value |
$1$ | $1$ | $()$ | $2$ |
$7$ | $2$ | $(1,7)(2,3)(4,6)$ | $0$ |
$2$ | $7$ | $(1,5,7,3,4,6,2)$ | $\zeta_{7}^{4} + \zeta_{7}^{3}$ |
$2$ | $7$ | $(1,7,4,2,5,3,6)$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$ |
$2$ | $7$ | $(1,3,2,7,6,5,4)$ | $\zeta_{7}^{5} + \zeta_{7}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.