Basic invariants
| Dimension: | $14$ |
| Group: | $S_7$ |
| Conductor: | \(676\!\cdots\!793\)\(\medspace = 563^{5} \cdot 103651^{5} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.7.58355513.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 30T565 |
| Parity: | even |
| Determinant: | 1.58355513.2t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.7.58355513.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 2x^{6} - 5x^{5} + 9x^{4} + 5x^{3} - 7x^{2} - x + 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 18 a + 5 + \left(16 a + 9\right)\cdot 19 + \left(15 a + 15\right)\cdot 19^{3} + 2\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 13 a + 5 + \left(7 a + 11\right)\cdot 19 + \left(13 a + 15\right)\cdot 19^{2} + \left(13 a + 5\right)\cdot 19^{3} + \left(a + 9\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 14 a + 1 + \left(3 a + 5\right)\cdot 19 + \left(10 a + 4\right)\cdot 19^{2} + \left(18 a + 3\right)\cdot 19^{3} + 8\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 11 + 3\cdot 19 + 19^{2} + 5\cdot 19^{3} + 3\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( a + 4 + \left(2 a + 8\right)\cdot 19 + \left(18 a + 3\right)\cdot 19^{2} + \left(3 a + 10\right)\cdot 19^{3} + \left(18 a + 7\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 6 a + 18 + \left(11 a + 5\right)\cdot 19 + \left(5 a + 2\right)\cdot 19^{2} + \left(5 a + 6\right)\cdot 19^{3} + \left(17 a + 16\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 5 a + 15 + \left(15 a + 13\right)\cdot 19 + \left(8 a + 10\right)\cdot 19^{2} + 11\cdot 19^{3} + \left(18 a + 9\right)\cdot 19^{4} +O(19^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $14$ |
| $21$ | $2$ | $(1,2)$ | $4$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $-1$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
| $210$ | $4$ | $(1,2,3,4)$ | $-2$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $-1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.