Basic invariants
Dimension: | $6$ |
Group: | $S_7$ |
Conductor: | \(383687\)\(\medspace = 31 \cdot 12377 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.1.383687.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_7$ |
Parity: | odd |
Determinant: | 1.383687.2t1.a.a |
Projective image: | $S_7$ |
Projective stem field: | Galois closure of 7.1.383687.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - x^{6} + 2x^{5} - 2x^{4} + x^{3} - x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$: \( x^{2} + 70x + 5 \)
Roots:
$r_{ 1 }$ | $=$ | \( 10 + 60\cdot 73 + 52\cdot 73^{2} + 34\cdot 73^{3} + 70\cdot 73^{4} +O(73^{5})\) |
$r_{ 2 }$ | $=$ | \( 34 + 33\cdot 73 + 71\cdot 73^{2} + 7\cdot 73^{3} + 73^{4} +O(73^{5})\) |
$r_{ 3 }$ | $=$ | \( 38 a + 66 + \left(63 a + 43\right)\cdot 73 + \left(49 a + 24\right)\cdot 73^{2} + \left(45 a + 60\right)\cdot 73^{3} + \left(54 a + 49\right)\cdot 73^{4} +O(73^{5})\) |
$r_{ 4 }$ | $=$ | \( 53 + 64\cdot 73 + 4\cdot 73^{2} + 73^{3} + 56\cdot 73^{4} +O(73^{5})\) |
$r_{ 5 }$ | $=$ | \( 35 a + 34 + \left(9 a + 50\right)\cdot 73 + \left(23 a + 37\right)\cdot 73^{2} + \left(27 a + 1\right)\cdot 73^{3} + \left(18 a + 22\right)\cdot 73^{4} +O(73^{5})\) |
$r_{ 6 }$ | $=$ | \( 33 a + 35 + \left(33 a + 22\right)\cdot 73 + \left(59 a + 14\right)\cdot 73^{2} + \left(53 a + 42\right)\cdot 73^{3} + \left(56 a + 24\right)\cdot 73^{4} +O(73^{5})\) |
$r_{ 7 }$ | $=$ | \( 40 a + 61 + \left(39 a + 16\right)\cdot 73 + \left(13 a + 13\right)\cdot 73^{2} + \left(19 a + 71\right)\cdot 73^{3} + \left(16 a + 67\right)\cdot 73^{4} +O(73^{5})\) |
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 |
$1$ | $1$ | $()$ | $6$ |
$21$ | $2$ | $(1,2)$ | $4$ |
$105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
$105$ | $2$ | $(1,2)(3,4)$ | $2$ |
$70$ | $3$ | $(1,2,3)$ | $3$ |
$280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
$210$ | $4$ | $(1,2,3,4)$ | $2$ |
$630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
$504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
$210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
$420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
$840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
$720$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ |
$504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $-1$ |
$420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.