Basic invariants
| Dimension: | $6$ |
| Group: | $S_7$ |
| Conductor: | \(146\!\cdots\!299\)\(\medspace = 4297259^{5} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 7.5.4297259.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 14T46 |
| Parity: | odd |
| Projective image: | $S_7$ |
| Projective field: | Galois closure of 7.5.4297259.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 157 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 157 }$:
\( x^{2} + 152x + 5 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 26 a + 149 + \left(50 a + 42\right)\cdot 157 + \left(60 a + 47\right)\cdot 157^{2} + \left(30 a + 51\right)\cdot 157^{3} + \left(114 a + 99\right)\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 22 + 78\cdot 157 + 104\cdot 157^{2} + 117\cdot 157^{3} + 85\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 63 a + 138 + \left(48 a + 98\right)\cdot 157 + \left(35 a + 102\right)\cdot 157^{2} + \left(94 a + 154\right)\cdot 157^{3} + \left(109 a + 4\right)\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 106 + 68\cdot 157 + 131\cdot 157^{2} + 145\cdot 157^{3} + 10\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 109 + 107\cdot 157 + 26\cdot 157^{2} + 53\cdot 157^{3} + 113\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 131 a + 122 + \left(106 a + 110\right)\cdot 157 + \left(96 a + 141\right)\cdot 157^{2} + \left(126 a + 142\right)\cdot 157^{3} + \left(42 a + 11\right)\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 94 a + 139 + \left(108 a + 120\right)\cdot 157 + \left(121 a + 73\right)\cdot 157^{2} + \left(62 a + 119\right)\cdot 157^{3} + \left(47 a + 144\right)\cdot 157^{4} +O(157^{5})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $6$ |
| $21$ | $2$ | $(1,2)$ | $-4$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $-2$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $1$ |