Basic invariants
| Dimension: | $6$ |
| Group: | $S_7$ |
| Conductor: | \(270607\)\(\medspace = 461 \cdot 587 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 7.1.270607.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_7$ |
| Parity: | odd |
| Projective image: | $S_7$ |
| Projective field: | Galois closure of 7.1.270607.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$:
\( x^{2} + 58x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 8\cdot 59 + 49\cdot 59^{2} + 43\cdot 59^{3} + 54\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 56 + 58\cdot 59 + 51\cdot 59^{2} + 24\cdot 59^{3} + 21\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 42 a + 14 + \left(29 a + 33\right)\cdot 59 + \left(29 a + 19\right)\cdot 59^{2} + \left(22 a + 6\right)\cdot 59^{3} + \left(14 a + 51\right)\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 54 + 29\cdot 59 + 11\cdot 59^{2} + 28\cdot 59^{3} + 44\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 17 a + 56 + \left(29 a + 20\right)\cdot 59 + \left(29 a + 19\right)\cdot 59^{2} + \left(36 a + 58\right)\cdot 59^{3} + \left(44 a + 42\right)\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 11 a + 22 + \left(8 a + 14\right)\cdot 59 + \left(22 a + 35\right)\cdot 59^{2} + \left(34 a + 1\right)\cdot 59^{3} + \left(57 a + 58\right)\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 48 a + 33 + \left(50 a + 11\right)\cdot 59 + \left(36 a + 49\right)\cdot 59^{2} + \left(24 a + 13\right)\cdot 59^{3} + \left(a + 22\right)\cdot 59^{4} +O(59^{5})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $6$ |
| $21$ | $2$ | $(1,2)$ | $4$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $2$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $-1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |