Basic invariants
Dimension: | $21$ |
Group: | $S_7$ |
Conductor: | \(332\!\cdots\!649\)\(\medspace = 61^{10} \cdot 4643^{10} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.1.283223.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 84 |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $S_7$ |
Projective stem field: | Galois closure of 7.1.283223.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} + x^{3} - 2x - 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 251 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 251 }$: \( x^{2} + 242x + 6 \)
Roots:
$r_{ 1 }$ | $=$ | \( 151 a + 29 + \left(204 a + 186\right)\cdot 251 + \left(33 a + 43\right)\cdot 251^{2} + \left(43 a + 66\right)\cdot 251^{3} + \left(127 a + 169\right)\cdot 251^{4} +O(251^{5})\) |
$r_{ 2 }$ | $=$ | \( 239 + 165\cdot 251 + 223\cdot 251^{2} + 181\cdot 251^{3} + 187\cdot 251^{4} +O(251^{5})\) |
$r_{ 3 }$ | $=$ | \( 247 a + 57 + \left(15 a + 75\right)\cdot 251 + \left(104 a + 24\right)\cdot 251^{2} + \left(81 a + 75\right)\cdot 251^{3} + \left(110 a + 186\right)\cdot 251^{4} +O(251^{5})\) |
$r_{ 4 }$ | $=$ | \( 132 + 145\cdot 251 + 242\cdot 251^{2} + 242\cdot 251^{3} + 26\cdot 251^{4} +O(251^{5})\) |
$r_{ 5 }$ | $=$ | \( 100 a + 133 + \left(46 a + 119\right)\cdot 251 + \left(217 a + 143\right)\cdot 251^{2} + \left(207 a + 169\right)\cdot 251^{3} + \left(123 a + 15\right)\cdot 251^{4} +O(251^{5})\) |
$r_{ 6 }$ | $=$ | \( 4 a + 21 + \left(235 a + 223\right)\cdot 251 + \left(146 a + 191\right)\cdot 251^{2} + \left(169 a + 201\right)\cdot 251^{3} + \left(140 a + 93\right)\cdot 251^{4} +O(251^{5})\) |
$r_{ 7 }$ | $=$ | \( 142 + 88\cdot 251 + 134\cdot 251^{2} + 66\cdot 251^{3} + 73\cdot 251^{4} +O(251^{5})\) |
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 |
$1$ | $1$ | $()$ | $21$ |
$21$ | $2$ | $(1,2)$ | $1$ |
$105$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
$105$ | $2$ | $(1,2)(3,4)$ | $1$ |
$70$ | $3$ | $(1,2,3)$ | $-3$ |
$280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
$210$ | $4$ | $(1,2,3,4)$ | $-1$ |
$630$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
$504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
$210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $1$ |
$420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
$840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
$720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
$504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $1$ |
$420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.