Basic invariants
Dimension: | $14$ |
Group: | $S_7$ |
Conductor: | \(254\!\cdots\!881\)\(\medspace = 277^{4} \cdot 811^{4} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.1.224647.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 21T38 |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $S_7$ |
Projective stem field: | Galois closure of 7.1.224647.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - 2x^{5} - x^{4} + 2x^{3} + 2x^{2} - 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 151 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 151 }$: \( x^{2} + 149x + 6 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 142 a + 8 + \left(17 a + 84\right)\cdot 151 + \left(49 a + 63\right)\cdot 151^{2} + \left(61 a + 45\right)\cdot 151^{3} + \left(140 a + 126\right)\cdot 151^{4} +O(151^{5})\)
$r_{ 2 }$ |
$=$ |
\( 10 + 138\cdot 151 + 38\cdot 151^{2} + 62\cdot 151^{3} + 91\cdot 151^{4} +O(151^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 95 a + 133 + \left(7 a + 97\right)\cdot 151 + \left(14 a + 53\right)\cdot 151^{2} + \left(93 a + 149\right)\cdot 151^{3} + \left(44 a + 107\right)\cdot 151^{4} +O(151^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 128 a + 93 + \left(143 a + 139\right)\cdot 151 + \left(44 a + 141\right)\cdot 151^{2} + \left(91 a + 110\right)\cdot 151^{3} + \left(35 a + 150\right)\cdot 151^{4} +O(151^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 56 a + 21 + \left(143 a + 18\right)\cdot 151 + \left(136 a + 74\right)\cdot 151^{2} + \left(57 a + 19\right)\cdot 151^{3} + \left(106 a + 104\right)\cdot 151^{4} +O(151^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 9 a + 141 + \left(133 a + 128\right)\cdot 151 + \left(101 a + 143\right)\cdot 151^{2} + \left(89 a + 118\right)\cdot 151^{3} + \left(10 a + 43\right)\cdot 151^{4} +O(151^{5})\)
| $r_{ 7 }$ |
$=$ |
\( 23 a + 47 + \left(7 a + 148\right)\cdot 151 + \left(106 a + 87\right)\cdot 151^{2} + \left(59 a + 97\right)\cdot 151^{3} + \left(115 a + 130\right)\cdot 151^{4} +O(151^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $14$ |
$21$ | $2$ | $(1,2)$ | $6$ |
$105$ | $2$ | $(1,2)(3,4)(5,6)$ | $2$ |
$105$ | $2$ | $(1,2)(3,4)$ | $2$ |
$70$ | $3$ | $(1,2,3)$ | $2$ |
$280$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
$210$ | $4$ | $(1,2,3,4)$ | $0$ |
$630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
$504$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
$210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $2$ |
$420$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
$840$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
$720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
$504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $1$ |
$420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.