Basic invariants
| Dimension: | $14$ |
| Group: | $S_7$ |
| Conductor: | \(983\!\cdots\!439\)\(\medspace = 89^{9} \cdot 67231^{9} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.5.5983559.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 30T565 |
| Parity: | odd |
| Determinant: | 1.5983559.2t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.5.5983559.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - x^{5} - x^{4} - 3x^{3} + 2x^{2} + 2x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 149 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 149 }$:
\( x^{2} + 145x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 9 a + 105 + \left(97 a + 6\right)\cdot 149 + \left(147 a + 94\right)\cdot 149^{2} + \left(110 a + 44\right)\cdot 149^{3} + \left(14 a + 8\right)\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 104 a + 8 + \left(41 a + 33\right)\cdot 149 + \left(25 a + 78\right)\cdot 149^{2} + \left(98 a + 133\right)\cdot 149^{3} + \left(97 a + 101\right)\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 95 + 30\cdot 149 + 148\cdot 149^{2} + 81\cdot 149^{3} + 57\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 45 a + 126 + \left(107 a + 95\right)\cdot 149 + \left(123 a + 137\right)\cdot 149^{2} + \left(50 a + 53\right)\cdot 149^{3} + \left(51 a + 96\right)\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 74 + 133\cdot 149 + 118\cdot 149^{2} + 12\cdot 149^{3} + 43\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 140 a + 141 + \left(51 a + 87\right)\cdot 149 + \left(a + 140\right)\cdot 149^{2} + \left(38 a + 42\right)\cdot 149^{3} + \left(134 a + 105\right)\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 47 + 59\cdot 149 + 27\cdot 149^{2} + 77\cdot 149^{3} + 34\cdot 149^{4} +O(149^{5})\)
|
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 value | Complex conjugation |
| $1$ | $1$ | $()$ | $14$ | |
| $21$ | $2$ | $(1,2)$ | $-4$ | ✓ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ | |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ | |
| $70$ | $3$ | $(1,2,3)$ | $-1$ | |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ | |
| $210$ | $4$ | $(1,2,3,4)$ | $2$ | |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ | |
| $504$ | $5$ | $(1,2,3,4,5)$ | $-1$ | |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ | |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ | |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ | |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ | |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $1$ | |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |