Basic invariants
| Dimension: | $2$ |
| Group: | $D_{7}$ |
| Conductor: | \(1859\)\(\medspace = 11 \cdot 13^{2}\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.1.6424482779.1 |
| Galois orbit size: | $3$ |
| Smallest permutation container: | $D_{7}$ |
| Parity: | odd |
| Determinant: | 1.11.2t1.a.a |
| Projective image: | $D_7$ |
| Projective stem field: | Galois closure of 7.1.6424482779.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 3x^{6} + 2x^{5} - 7x^{4} + 42x^{3} - 3x^{2} - 267x + 344 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 15 a + 9 + \left(13 a + 14\right)\cdot 17 + 5\cdot 17^{2} + \left(5 a + 13\right)\cdot 17^{3} + \left(a + 13\right)\cdot 17^{4} +O(17^{5})\)
| $r_{ 2 }$ |
$=$ |
\( 15 a + 10 + 14 a\cdot 17 + 15 a\cdot 17^{2} + \left(12 a + 16\right)\cdot 17^{3} + \left(a + 13\right)\cdot 17^{4} +O(17^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 15 + 3\cdot 17 + 4\cdot 17^{2} + 7\cdot 17^{3} + 2\cdot 17^{4} +O(17^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 2 a + 8 + 2 a\cdot 17 + \left(a + 1\right)\cdot 17^{2} + \left(4 a + 13\right)\cdot 17^{3} + \left(15 a + 2\right)\cdot 17^{4} +O(17^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 11 a + 14 + \left(11 a + 8\right)\cdot 17 + 3\cdot 17^{2} + \left(11 a + 12\right)\cdot 17^{3} + \left(12 a + 11\right)\cdot 17^{4} +O(17^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 2 a + 7 + \left(3 a + 13\right)\cdot 17 + \left(16 a + 9\right)\cdot 17^{2} + 11 a\cdot 17^{3} + \left(15 a + 10\right)\cdot 17^{4} +O(17^{5})\)
| $r_{ 7 }$ |
$=$ |
\( 6 a + 8 + \left(5 a + 9\right)\cdot 17 + \left(16 a + 9\right)\cdot 17^{2} + \left(5 a + 5\right)\cdot 17^{3} + \left(4 a + 13\right)\cdot 17^{4} +O(17^{5})\)
| |
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 |
| $1$ | $1$ | $()$ | $2$ |
| $7$ | $2$ | $(1,5)(3,6)(4,7)$ | $0$ |
| $2$ | $7$ | $(1,2,5,3,7,4,6)$ | $\zeta_{7}^{4} + \zeta_{7}^{3}$ |
| $2$ | $7$ | $(1,5,7,6,2,3,4)$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$ |
| $2$ | $7$ | $(1,3,6,5,4,2,7)$ | $\zeta_{7}^{5} + \zeta_{7}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.