Basic invariants
| Dimension: | $2$ |
| Group: | $D_{7}$ |
| Conductor: | \(536\)\(\medspace = 2^{3} \cdot 67 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.1.153990656.1 |
| Galois orbit size: | $3$ |
| Smallest permutation container: | $D_{7}$ |
| Parity: | odd |
| Determinant: | 1.536.2t1.b.a |
| Projective image: | $D_7$ |
| Projective stem field: | Galois closure of 7.1.153990656.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 2x^{6} + 2x^{5} - x^{3} - 6x^{2} - 6x - 4 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{2} + 18x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 17 a + 7 + 9 a\cdot 19 + \left(7 a + 11\right)\cdot 19^{2} + \left(2 a + 3\right)\cdot 19^{3} + \left(9 a + 4\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 16 a + 18 + \left(14 a + 9\right)\cdot 19 + \left(17 a + 18\right)\cdot 19^{2} + 18 a\cdot 19^{3} + \left(17 a + 4\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 6 + 6\cdot 19 + 8\cdot 19^{2} + 11\cdot 19^{3} + 6\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 13 a + 7 + \left(10 a + 1\right)\cdot 19 + \left(5 a + 16\right)\cdot 19^{2} + 3\cdot 19^{3} + \left(12 a + 8\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 2 a + 5 + \left(9 a + 12\right)\cdot 19 + \left(11 a + 8\right)\cdot 19^{2} + \left(16 a + 17\right)\cdot 19^{3} + \left(9 a + 10\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 6 a + 1 + \left(8 a + 18\right)\cdot 19 + \left(13 a + 10\right)\cdot 19^{2} + \left(18 a + 17\right)\cdot 19^{3} + 6 a\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 3 a + 15 + \left(4 a + 8\right)\cdot 19 + \left(a + 2\right)\cdot 19^{2} + 2\cdot 19^{3} + \left(a + 3\right)\cdot 19^{4} +O(19^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $7$ | $2$ | $(1,7)(2,6)(3,4)$ | $0$ |
| $2$ | $7$ | $(1,3,5,4,7,2,6)$ | $\zeta_{7}^{4} + \zeta_{7}^{3}$ |
| $2$ | $7$ | $(1,5,7,6,3,4,2)$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$ |
| $2$ | $7$ | $(1,4,6,5,2,3,7)$ | $\zeta_{7}^{5} + \zeta_{7}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.