Properties

Label 6.315631.7t7.a
Dimension $6$
Group $S_7$
Conductor $315631$
Indicator $1$

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Basic invariants

Dimension:$6$
Group:$S_7$
Conductor:\(315631\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 7.1.315631.1
Galois orbit size: $1$
Smallest permutation container: $S_7$
Parity: odd
Projective image: $S_7$
Projective field: Galois closure of 7.1.315631.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 163 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 163 }$: \( x^{2} + 159x + 2 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 46 + 120\cdot 163 + 63\cdot 163^{2} + 62\cdot 163^{3} + 5\cdot 163^{4} +O(163^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 100 a + 25 + \left(71 a + 81\right)\cdot 163 + \left(39 a + 131\right)\cdot 163^{2} + \left(2 a + 129\right)\cdot 163^{3} + \left(66 a + 60\right)\cdot 163^{4} +O(163^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 38 a + 23 + \left(162 a + 69\right)\cdot 163 + \left(76 a + 150\right)\cdot 163^{2} + \left(49 a + 65\right)\cdot 163^{3} + \left(132 a + 143\right)\cdot 163^{4} +O(163^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 109 + 86\cdot 163 + 9\cdot 163^{2} + 100\cdot 163^{3} + 126\cdot 163^{4} +O(163^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 12 + 162\cdot 163 + 108\cdot 163^{2} + 7\cdot 163^{3} + 22\cdot 163^{4} +O(163^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 125 a + 12 + 28\cdot 163 + \left(86 a + 133\right)\cdot 163^{2} + \left(113 a + 23\right)\cdot 163^{3} + \left(30 a + 134\right)\cdot 163^{4} +O(163^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 63 a + 99 + \left(91 a + 104\right)\cdot 163 + \left(123 a + 54\right)\cdot 163^{2} + \left(160 a + 99\right)\cdot 163^{3} + \left(96 a + 159\right)\cdot 163^{4} +O(163^{5})\) Copy content Toggle raw display

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.