Basic invariants
Dimension: | $2$ |
Group: | $D_{7}$ |
Conductor: | \(3743\)\(\medspace = 19 \cdot 197 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.1.52439613407.1 |
Galois orbit size: | $3$ |
Smallest permutation container: | $D_{7}$ |
Parity: | odd |
Determinant: | 1.3743.2t1.a.a |
Projective image: | $D_7$ |
Projective stem field: | Galois closure of 7.1.52439613407.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - 2x^{6} + 11x^{5} - 26x^{4} + 45x^{3} - 60x^{2} + 39x + 11 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \( x^{2} + 16x + 3 \)
Roots:
$r_{ 1 }$ | $=$ | \( 3 a + \left(9 a + 4\right)\cdot 17 + \left(6 a + 8\right)\cdot 17^{2} + \left(2 a + 16\right)\cdot 17^{3} + \left(12 a + 8\right)\cdot 17^{4} +O(17^{5})\) |
$r_{ 2 }$ | $=$ | \( 14 a + 3 + \left(7 a + 10\right)\cdot 17 + \left(10 a + 5\right)\cdot 17^{2} + \left(14 a + 12\right)\cdot 17^{3} + \left(4 a + 1\right)\cdot 17^{4} +O(17^{5})\) |
$r_{ 3 }$ | $=$ | \( 4 a + 12 + \left(9 a + 13\right)\cdot 17 + \left(7 a + 8\right)\cdot 17^{2} + \left(7 a + 15\right)\cdot 17^{3} + \left(2 a + 14\right)\cdot 17^{4} +O(17^{5})\) |
$r_{ 4 }$ | $=$ | \( 14 + 11\cdot 17 + 3\cdot 17^{2} + 6\cdot 17^{3} + 11\cdot 17^{4} +O(17^{5})\) |
$r_{ 5 }$ | $=$ | \( 13 a + 16 + \left(7 a + 1\right)\cdot 17 + \left(9 a + 7\right)\cdot 17^{2} + \left(9 a + 15\right)\cdot 17^{3} + \left(14 a + 9\right)\cdot 17^{4} +O(17^{5})\) |
$r_{ 6 }$ | $=$ | \( 8 a + \left(8 a + 13\right)\cdot 17 + \left(14 a + 5\right)\cdot 17^{2} + \left(14 a + 9\right)\cdot 17^{3} + \left(14 a + 10\right)\cdot 17^{4} +O(17^{5})\) |
$r_{ 7 }$ | $=$ | \( 9 a + 8 + \left(8 a + 13\right)\cdot 17 + \left(2 a + 11\right)\cdot 17^{2} + \left(2 a + 9\right)\cdot 17^{3} + \left(2 a + 10\right)\cdot 17^{4} +O(17^{5})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$7$ | $2$ | $(1,7)(2,4)(3,6)$ | $0$ |
$2$ | $7$ | $(1,4,2,7,3,5,6)$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$ |
$2$ | $7$ | $(1,2,3,6,4,7,5)$ | $\zeta_{7}^{5} + \zeta_{7}^{2}$ |
$2$ | $7$ | $(1,7,6,2,5,4,3)$ | $\zeta_{7}^{4} + \zeta_{7}^{3}$ |
The blue line marks the conjugacy class containing complex conjugation.