Basic invariants
| Dimension: | $6$ |
| Group: | $S_7$ |
| Conductor: | \(250367\)\(\medspace = 13 \cdot 19259 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 7.1.250367.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_7$ |
| Parity: | odd |
| Projective image: | $S_7$ |
| Projective field: | Galois closure of 7.1.250367.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$:
\( x^{2} + 97x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 18 a + 14 + \left(24 a + 17\right)\cdot 101 + \left(82 a + 3\right)\cdot 101^{2} + \left(12 a + 63\right)\cdot 101^{3} + \left(69 a + 59\right)\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 74 a + 31 + \left(54 a + 26\right)\cdot 101 + \left(70 a + 80\right)\cdot 101^{2} + \left(16 a + 100\right)\cdot 101^{3} + 24\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 2 a + 87 + \left(43 a + 68\right)\cdot 101 + \left(17 a + 44\right)\cdot 101^{2} + \left(9 a + 95\right)\cdot 101^{3} + \left(34 a + 91\right)\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 83 a + 86 + \left(76 a + 95\right)\cdot 101 + \left(18 a + 4\right)\cdot 101^{2} + \left(88 a + 32\right)\cdot 101^{3} + \left(31 a + 20\right)\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 69 + 88\cdot 101 + 93\cdot 101^{2} + 101^{3} + 80\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 27 a + 24 + \left(46 a + 70\right)\cdot 101 + \left(30 a + 4\right)\cdot 101^{2} + \left(84 a + 97\right)\cdot 101^{3} + \left(100 a + 8\right)\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 99 a + 95 + \left(57 a + 36\right)\cdot 101 + \left(83 a + 71\right)\cdot 101^{2} + \left(91 a + 13\right)\cdot 101^{3} + \left(66 a + 17\right)\cdot 101^{4} +O(101^{5})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $6$ |
| $21$ | $2$ | $(1,2)$ | $4$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $2$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $-1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |