Properties

Label 6.184607.7t7.1c1
Dimension 6
Group $S_7$
Conductor $ 184607 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$6$
Group:$S_7$
Conductor:$184607 $
Artin number field: Splitting field of $f= x^{7} - x^{6} - x^{5} + x^{4} - x^{2} + x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_7$
Parity: Odd
Determinant: 1.184607.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 103 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 103 }$: $ x^{2} + 102 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 39 + 12\cdot 103 + 56\cdot 103^{2} + 59\cdot 103^{3} + 33\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 86 + 6\cdot 103 + 91\cdot 103^{2} + 65\cdot 103^{3} + 35\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 80 a + 78 + \left(100 a + 73\right)\cdot 103 + \left(57 a + 89\right)\cdot 103^{2} + \left(11 a + 79\right)\cdot 103^{3} + \left(19 a + 31\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 4 a + 14 + \left(22 a + 94\right)\cdot 103 + \left(53 a + 72\right)\cdot 103^{2} + \left(57 a + 83\right)\cdot 103^{3} + \left(97 a + 9\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 99 a + 18 + \left(80 a + 9\right)\cdot 103 + \left(49 a + 1\right)\cdot 103^{2} + \left(45 a + 88\right)\cdot 103^{3} + \left(5 a + 49\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 23 a + 55 + \left(2 a + 94\right)\cdot 103 + \left(45 a + 46\right)\cdot 103^{2} + \left(91 a + 33\right)\cdot 103^{3} + \left(83 a + 39\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 20 + 18\cdot 103 + 54\cdot 103^{2} + 103^{3} + 6\cdot 103^{4} +O\left(103^{ 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.