Basic invariants
| Dimension: | $2$ |
| Group: | $D_{7}$ |
| Conductor: | \(391\)\(\medspace = 17 \cdot 23 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 7.1.59776471.1 |
| Galois orbit size: | $3$ |
| Smallest permutation container: | $D_{7}$ |
| Parity: | odd |
| Projective image: | $D_7$ |
| Projective field: | Galois closure of 7.1.59776471.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{2} + 18x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( a + 17 + \left(8 a + 4\right)\cdot 19 + \left(12 a + 18\right)\cdot 19^{2} + \left(3 a + 9\right)\cdot 19^{3} + \left(16 a + 18\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 7 a + 9 + \left(4 a + 3\right)\cdot 19 + \left(3 a + 4\right)\cdot 19^{2} + \left(10 a + 6\right)\cdot 19^{3} + \left(15 a + 9\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 18 a + 18 + \left(10 a + 11\right)\cdot 19 + \left(6 a + 3\right)\cdot 19^{2} + \left(15 a + 1\right)\cdot 19^{3} + \left(2 a + 12\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 3 a + 4 + \left(7 a + 3\right)\cdot 19 + \left(12 a + 13\right)\cdot 19^{2} + 2\cdot 19^{3} + \left(14 a + 7\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 7 + 6\cdot 19 + 15\cdot 19^{2} + 13\cdot 19^{3} + 12\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 12 a + 16 + 14 a\cdot 19 + \left(15 a + 3\right)\cdot 19^{2} + \left(8 a + 13\right)\cdot 19^{3} + \left(3 a + 14\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 16 a + 7 + \left(11 a + 7\right)\cdot 19 + \left(6 a + 18\right)\cdot 19^{2} + \left(18 a + 9\right)\cdot 19^{3} + \left(4 a + 1\right)\cdot 19^{4} +O(19^{5})\)
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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 values | ||
| $c1$ | $c2$ | $c3$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ | $2$ |
| $7$ | $2$ | $(1,7)(2,3)(4,5)$ | $0$ | $0$ | $0$ |
| $2$ | $7$ | $(1,5,3,2,4,7,6)$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$ | $\zeta_{7}^{4} + \zeta_{7}^{3}$ | $\zeta_{7}^{5} + \zeta_{7}^{2}$ |
| $2$ | $7$ | $(1,3,4,6,5,2,7)$ | $\zeta_{7}^{5} + \zeta_{7}^{2}$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$ | $\zeta_{7}^{4} + \zeta_{7}^{3}$ |
| $2$ | $7$ | $(1,2,6,3,7,5,4)$ | $\zeta_{7}^{4} + \zeta_{7}^{3}$ | $\zeta_{7}^{5} + \zeta_{7}^{2}$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$ |