Basic invariants
| Dimension: | $14$ |
| Group: | $S_7$ |
| Conductor: | \(844\!\cdots\!873\)\(\medspace = 35269513^{9} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.7.35269513.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 30T565 |
| Parity: | even |
| Determinant: | 1.35269513.2t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.7.35269513.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 2x^{6} - 5x^{5} + 10x^{4} + 5x^{3} - 10x^{2} - x + 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 120 a + 78 + \left(116 a + 116\right)\cdot 149 + \left(95 a + 80\right)\cdot 149^{2} + \left(129 a + 45\right)\cdot 149^{3} + 106 a\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 142 a + 15 + \left(80 a + 10\right)\cdot 149 + \left(53 a + 132\right)\cdot 149^{2} + \left(54 a + 128\right)\cdot 149^{3} + \left(131 a + 144\right)\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 7 a + 136 + \left(68 a + 42\right)\cdot 149 + \left(95 a + 116\right)\cdot 149^{2} + \left(94 a + 143\right)\cdot 149^{3} + \left(17 a + 19\right)\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 29 a + 111 + \left(32 a + 16\right)\cdot 149 + \left(53 a + 49\right)\cdot 149^{2} + \left(19 a + 21\right)\cdot 149^{3} + 42 a\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 20 a + 21 + \left(25 a + 128\right)\cdot 149 + \left(126 a + 9\right)\cdot 149^{2} + \left(78 a + 127\right)\cdot 149^{3} + \left(31 a + 100\right)\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 129 a + 101 + \left(123 a + 59\right)\cdot 149 + \left(22 a + 42\right)\cdot 149^{2} + \left(70 a + 18\right)\cdot 149^{3} + \left(117 a + 148\right)\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 136 + 72\cdot 149 + 16\cdot 149^{2} + 111\cdot 149^{3} + 32\cdot 149^{4} +O(149^{5})\)
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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 |
| $1$ | $1$ | $()$ | $14$ | ✓ |
| $21$ | $2$ | $(1,2)$ | $-4$ | |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ | |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ | |
| $70$ | $3$ | $(1,2,3)$ | $-1$ | |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ | |
| $210$ | $4$ | $(1,2,3,4)$ | $2$ | |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ | |
| $504$ | $5$ | $(1,2,3,4,5)$ | $-1$ | |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ | |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ | |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ | |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ | |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $1$ | |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |