Properties

Label 6.332447.7t7.a.a
Dimension $6$
Group $S_7$
Conductor $332447$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $6$
Group: $S_7$
Conductor: \(332447\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.332447.1
Galois orbit size: $1$
Smallest permutation container: $S_7$
Parity: odd
Determinant: 1.332447.2t1.a.a
Projective image: $S_7$
Projective stem field: Galois closure of 7.1.332447.1

Defining polynomial

$f(x)$$=$ \( x^{7} - x^{5} - x^{4} + x^{3} + x^{2} - x - 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 179 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 179 }$: \( x^{2} + 172x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 137 + 43\cdot 179 + 4\cdot 179^{2} + 66\cdot 179^{3} + 59\cdot 179^{4} +O(179^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 132 a + 166 + \left(92 a + 17\right)\cdot 179 + \left(166 a + 80\right)\cdot 179^{2} + \left(62 a + 103\right)\cdot 179^{3} + \left(150 a + 20\right)\cdot 179^{4} +O(179^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 162 + 68\cdot 179 + 31\cdot 179^{2} + 3\cdot 179^{3} + 138\cdot 179^{4} +O(179^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 52 + 165\cdot 179 + 138\cdot 179^{2} + 95\cdot 179^{3} + 121\cdot 179^{4} +O(179^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 47 a + 16 + \left(86 a + 177\right)\cdot 179 + \left(12 a + 78\right)\cdot 179^{2} + \left(116 a + 19\right)\cdot 179^{3} + \left(28 a + 115\right)\cdot 179^{4} +O(179^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 51 a + 92 + \left(117 a + 94\right)\cdot 179 + \left(47 a + 172\right)\cdot 179^{2} + \left(39 a + 10\right)\cdot 179^{3} + \left(122 a + 170\right)\cdot 179^{4} +O(179^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 128 a + 91 + \left(61 a + 148\right)\cdot 179 + \left(131 a + 30\right)\cdot 179^{2} + \left(139 a + 59\right)\cdot 179^{3} + \left(56 a + 91\right)\cdot 179^{4} +O(179^{5})\) 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 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.