Properties

Label 6.332447.7t7.1c1
Dimension 6
Group $S_7$
Conductor $ 332447 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$6$
Group:$S_7$
Conductor:$332447 $
Artin number field: Splitting field of $f= x^{7} - x^{5} - x^{4} + x^{3} + x^{2} - x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_7$
Parity: Odd
Determinant: 1.332447.2t1.1c1

Galois action

Roots of defining polynomial

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} + 172 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 137 + 43\cdot 179 + 4\cdot 179^{2} + 66\cdot 179^{3} + 59\cdot 179^{4} +O\left(179^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 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\left(179^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 162 + 68\cdot 179 + 31\cdot 179^{2} + 3\cdot 179^{3} + 138\cdot 179^{4} +O\left(179^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 52 + 165\cdot 179 + 138\cdot 179^{2} + 95\cdot 179^{3} + 121\cdot 179^{4} +O\left(179^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 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\left(179^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 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\left(179^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 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\left(179^{ 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 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.