Properties

Label 6.227287.7t7.a
Dimension 6
Group $S_7$
Conductor $ 167 \cdot 1361 $
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$6$
Group:$S_7$
Conductor:$227287= 167 \cdot 1361 $
Artin number field: Splitting field of $f= x^{7} - 2 x^{6} + 3 x^{5} - 3 x^{4} + 3 x^{3} - 2 x^{2} + 2 x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_7$
Parity: Odd
Projective image: $S_7$
Projective field: Galois closure of 7.1.227287.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $ x^{2} + 18 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 17 a + 16 + \left(17 a + 8\right)\cdot 19 + \left(11 a + 6\right)\cdot 19^{2} + \left(7 a + 18\right)\cdot 19^{3} + \left(14 a + 5\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 17 + 16\cdot 19 + 16\cdot 19^{2} + 19^{3} + 13\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 5 a + 1 + \left(3 a + 10\right)\cdot 19 + \left(17 a + 1\right)\cdot 19^{2} + \left(3 a + 15\right)\cdot 19^{3} + \left(18 a + 14\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 2 a + 14 + \left(a + 9\right)\cdot 19 + 7 a\cdot 19^{2} + \left(11 a + 14\right)\cdot 19^{3} + \left(4 a + 12\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 5 a + \left(10 a + 18\right)\cdot 19 + \left(15 a + 14\right)\cdot 19^{2} + \left(18 a + 10\right)\cdot 19^{3} + \left(6 a + 15\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 14 a + 6 + \left(15 a + 8\right)\cdot 19 + \left(a + 15\right)\cdot 19^{2} + \left(15 a + 1\right)\cdot 19^{3} + 10\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 14 a + 5 + \left(8 a + 4\right)\cdot 19 + \left(3 a + 1\right)\cdot 19^{2} + 14\cdot 19^{3} + \left(12 a + 3\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 7 }$

Cycle notation
$(1,2,3,4,5,6,7)$
$(1,2)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 7 }$ Character values
$c1$
$1$ $1$ $()$ $6$
$21$ $2$ $(1,2)$ $4$
$105$ $2$ $(1,2)(3,4)(5,6)$ $0$
$105$ $2$ $(1,2)(3,4)$ $2$
$70$ $3$ $(1,2,3)$ $3$
$280$ $3$ $(1,2,3)(4,5,6)$ $0$
$210$ $4$ $(1,2,3,4)$ $2$
$630$ $4$ $(1,2,3,4)(5,6)$ $0$
$504$ $5$ $(1,2,3,4,5)$ $1$
$210$ $6$ $(1,2,3)(4,5)(6,7)$ $-1$
$420$ $6$ $(1,2,3)(4,5)$ $1$
$840$ $6$ $(1,2,3,4,5,6)$ $0$
$720$ $7$ $(1,2,3,4,5,6,7)$ $-1$
$504$ $10$ $(1,2,3,4,5)(6,7)$ $-1$
$420$ $12$ $(1,2,3,4)(5,6,7)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.