Basic invariants
| Dimension: | $6$ |
| Group: | $S_7$ |
| Conductor: | \(332447\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 7.1.332447.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_7$ |
| Parity: | odd |
| Projective image: | $S_7$ |
| Projective field: | Galois closure of 7.1.332447.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 179 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 179 }$:
\( x^{2} + 172x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 137 + 43\cdot 179 + 4\cdot 179^{2} + 66\cdot 179^{3} + 59\cdot 179^{4} +O(179^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 132 a + 166 + \left(92 a + 17\right)\cdot 179 + \left(166 a + 80\right)\cdot 179^{2} + \left(62 a + 103\right)\cdot 179^{3} + \left(150 a + 20\right)\cdot 179^{4} +O(179^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 162 + 68\cdot 179 + 31\cdot 179^{2} + 3\cdot 179^{3} + 138\cdot 179^{4} +O(179^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 52 + 165\cdot 179 + 138\cdot 179^{2} + 95\cdot 179^{3} + 121\cdot 179^{4} +O(179^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 47 a + 16 + \left(86 a + 177\right)\cdot 179 + \left(12 a + 78\right)\cdot 179^{2} + \left(116 a + 19\right)\cdot 179^{3} + \left(28 a + 115\right)\cdot 179^{4} +O(179^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 51 a + 92 + \left(117 a + 94\right)\cdot 179 + \left(47 a + 172\right)\cdot 179^{2} + \left(39 a + 10\right)\cdot 179^{3} + \left(122 a + 170\right)\cdot 179^{4} +O(179^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 128 a + 91 + \left(61 a + 148\right)\cdot 179 + \left(131 a + 30\right)\cdot 179^{2} + \left(139 a + 59\right)\cdot 179^{3} + \left(56 a + 91\right)\cdot 179^{4} +O(179^{5})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $6$ |
| $21$ | $2$ | $(1,2)$ | $4$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $2$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $-1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |