Basic invariants
| Dimension: | $6$ |
| Group: | $S_7$ |
| Conductor: | \(594751\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 7.1.594751.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_7$ |
| Parity: | odd |
| Projective image: | $S_7$ |
| Projective field: | Galois closure of 7.1.594751.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 127 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 127 }$:
\( x^{2} + 126x + 3 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 84 + 93\cdot 127 + 126\cdot 127^{2} + 92\cdot 127^{3} + 36\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 38 + 35\cdot 127 + 40\cdot 127^{2} + 63\cdot 127^{3} + 116\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 14 a + 91 + \left(31 a + 45\right)\cdot 127 + \left(27 a + 88\right)\cdot 127^{2} + \left(87 a + 99\right)\cdot 127^{3} + \left(68 a + 25\right)\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 68 + 85\cdot 127 + 51\cdot 127^{2} + 41\cdot 127^{3} + 3\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 113 a + 105 + \left(95 a + 62\right)\cdot 127 + \left(99 a + 84\right)\cdot 127^{2} + \left(39 a + 32\right)\cdot 127^{3} + \left(58 a + 7\right)\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 53 a + 35 + \left(50 a + 30\right)\cdot 127 + \left(77 a + 108\right)\cdot 127^{2} + \left(105 a + 74\right)\cdot 127^{3} + \left(52 a + 58\right)\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 74 a + 88 + \left(76 a + 27\right)\cdot 127 + \left(49 a + 8\right)\cdot 127^{2} + \left(21 a + 103\right)\cdot 127^{3} + \left(74 a + 5\right)\cdot 127^{4} +O(127^{5})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $6$ |
| $21$ | $2$ | $(1,2)$ | $4$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $2$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $-1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |