Basic invariants
| Dimension: | $2$ |
| Group: | $D_{7}$ |
| Conductor: | \(3143\)\(\medspace = 7 \cdot 449 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.1.31047965207.1 |
| Galois orbit size: | $3$ |
| Smallest permutation container: | $D_{7}$ |
| Parity: | odd |
| Determinant: | 1.3143.2t1.a.a |
| Projective image: | $D_7$ |
| Projective stem field: | Galois closure of 7.1.31047965207.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 3x^{6} - 4x^{5} + 30x^{4} - 24x^{3} - 70x^{2} + 112x - 47 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 11 a + 6 + 10\cdot 29 + 9\cdot 29^{2} + \left(23 a + 7\right)\cdot 29^{3} + \left(13 a + 13\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 19 a + 9 + \left(6 a + 14\right)\cdot 29 + \left(19 a + 22\right)\cdot 29^{2} + \left(28 a + 22\right)\cdot 29^{3} + \left(7 a + 18\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( a + 8 + \left(27 a + 10\right)\cdot 29 + \left(19 a + 20\right)\cdot 29^{2} + \left(27 a + 10\right)\cdot 29^{3} + \left(13 a + 16\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 18 a + 3 + \left(28 a + 1\right)\cdot 29 + \left(28 a + 9\right)\cdot 29^{2} + \left(5 a + 6\right)\cdot 29^{3} + \left(15 a + 1\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 10 a + 17 + \left(22 a + 28\right)\cdot 29 + \left(9 a + 24\right)\cdot 29^{2} + 29^{3} + \left(21 a + 1\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 5 + 23\cdot 29 + 23\cdot 29^{2} + 24\cdot 29^{3} + 6\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 28 a + 13 + \left(a + 28\right)\cdot 29 + \left(9 a + 5\right)\cdot 29^{2} + \left(a + 13\right)\cdot 29^{3} + 15 a\cdot 29^{4} +O(29^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $7$ | $2$ | $(1,5)(2,7)(3,6)$ | $0$ |
| $2$ | $7$ | $(1,4,5,7,6,3,2)$ | $\zeta_{7}^{5} + \zeta_{7}^{2}$ |
| $2$ | $7$ | $(1,5,6,2,4,7,3)$ | $\zeta_{7}^{4} + \zeta_{7}^{3}$ |
| $2$ | $7$ | $(1,7,2,5,3,4,6)$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.