Basic invariants
| Dimension: | $6$ |
| Group: | $S_7$ |
| Conductor: | \(338191\)\(\medspace = 7 \cdot 48313 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 7.1.338191.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_7$ |
| Parity: | odd |
| Projective image: | $S_7$ |
| Projective field: | Galois closure of 7.1.338191.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 113 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 113 }$:
\( x^{2} + 101x + 3 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 92 a + 87 + \left(91 a + 82\right)\cdot 113 + \left(21 a + 110\right)\cdot 113^{2} + \left(101 a + 14\right)\cdot 113^{3} + \left(26 a + 34\right)\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 21 a + 61 + \left(21 a + 75\right)\cdot 113 + \left(91 a + 54\right)\cdot 113^{2} + \left(11 a + 77\right)\cdot 113^{3} + \left(86 a + 29\right)\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 76 + 13\cdot 113 + 46\cdot 113^{2} + 81\cdot 113^{3} + 68\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 24 a + 12 + \left(5 a + 33\right)\cdot 113 + \left(22 a + 41\right)\cdot 113^{2} + \left(3 a + 16\right)\cdot 113^{3} + \left(9 a + 52\right)\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 72 + 90\cdot 113 + 77\cdot 113^{2} + 12\cdot 113^{3} +O(113^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 70 + 84\cdot 113 + 46\cdot 113^{2} + 103\cdot 113^{3} + 109\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 89 a + 74 + \left(107 a + 71\right)\cdot 113 + \left(90 a + 74\right)\cdot 113^{2} + \left(109 a + 32\right)\cdot 113^{3} + \left(103 a + 44\right)\cdot 113^{4} +O(113^{5})\)
|
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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $6$ |
| $21$ | $2$ | $(1,2)$ | $4$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $2$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $-1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |