Basic invariants
| Dimension: | $2$ |
| Group: | $D_{7}$ |
| Conductor: | \(343\)\(\medspace = 7^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.1.40353607.1 |
| Galois orbit size: | $3$ |
| Smallest permutation container: | $D_{7}$ |
| Parity: | odd |
| Determinant: | 1.7.2t1.a.a |
| Projective image: | $D_7$ |
| Projective stem field: | Galois closure of 7.1.40353607.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} + 7x^{3} - 7x^{2} + 7x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$:
\( x^{2} + 12x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 a + 6 + \left(9 a + 2\right)\cdot 13 + \left(8 a + 8\right)\cdot 13^{2} + \left(10 a + 12\right)\cdot 13^{3} + \left(8 a + 10\right)\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 8 + 9\cdot 13 + 13^{2} + 8\cdot 13^{3} + 5\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 5 a + 10 + \left(11 a + 3\right)\cdot 13 + \left(5 a + 2\right)\cdot 13^{2} + 10 a\cdot 13^{3} + 13^{4} +O(13^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 8 a + 2 + \left(a + 10\right)\cdot 13 + \left(7 a + 9\right)\cdot 13^{2} + \left(2 a + 4\right)\cdot 13^{3} + \left(12 a + 4\right)\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 3 a + 11 + \left(a + 11\right)\cdot 13 + \left(a + 4\right)\cdot 13^{2} + \left(2 a + 5\right)\cdot 13^{3} + \left(4 a + 9\right)\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 10 a + 1 + \left(11 a + 10\right)\cdot 13 + \left(11 a + 4\right)\cdot 13^{2} + \left(10 a + 6\right)\cdot 13^{3} + \left(8 a + 11\right)\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 5 a + 1 + \left(3 a + 4\right)\cdot 13 + \left(4 a + 7\right)\cdot 13^{2} + \left(2 a + 1\right)\cdot 13^{3} + \left(4 a + 9\right)\cdot 13^{4} +O(13^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $7$ | $2$ | $(1,4)(2,7)(3,6)$ | $0$ |
| $2$ | $7$ | $(1,5,4,2,3,6,7)$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$ |
| $2$ | $7$ | $(1,4,3,7,5,2,6)$ | $\zeta_{7}^{5} + \zeta_{7}^{2}$ |
| $2$ | $7$ | $(1,2,7,4,6,5,3)$ | $\zeta_{7}^{4} + \zeta_{7}^{3}$ |
The blue line marks the conjugacy class containing complex conjugation.