Basic invariants
| Dimension: | $6$ |
| Group: | $S_7$ |
| Conductor: | \(214607\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 7.1.214607.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_7$ |
| Parity: | odd |
| Projective image: | $S_7$ |
| Projective field: | Galois closure of 7.1.214607.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 24 a + 3 + \left(4 a + 14\right)\cdot 31 + \left(24 a + 15\right)\cdot 31^{2} + \left(18 a + 6\right)\cdot 31^{3} + \left(11 a + 30\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 6 a + 27 + \left(4 a + 27\right)\cdot 31 + \left(28 a + 27\right)\cdot 31^{2} + \left(24 a + 13\right)\cdot 31^{3} + \left(25 a + 4\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 22 a + 25 + \left(27 a + 12\right)\cdot 31 + \left(8 a + 12\right)\cdot 31^{2} + \left(12 a + 11\right)\cdot 31^{3} + \left(24 a + 23\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 9 a + 7 + \left(3 a + 15\right)\cdot 31 + \left(22 a + 2\right)\cdot 31^{2} + \left(18 a + 27\right)\cdot 31^{3} + \left(6 a + 28\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 4 + 24\cdot 31 + 19\cdot 31^{2} + 9\cdot 31^{3} + 2\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 25 a + 8 + \left(26 a + 30\right)\cdot 31 + \left(2 a + 17\right)\cdot 31^{2} + \left(6 a + 4\right)\cdot 31^{3} + 5 a\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 7 a + 20 + \left(26 a + 30\right)\cdot 31 + \left(6 a + 27\right)\cdot 31^{2} + \left(12 a + 19\right)\cdot 31^{3} + \left(19 a + 3\right)\cdot 31^{4} +O(31^{5})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $6$ |
| $21$ | $2$ | $(1,2)$ | $4$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $2$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $-1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |