Properties

Label 6.382631.7t7.1c1
Dimension 6
Group $S_7$
Conductor $ 382631 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$6$
Group:$S_7$
Conductor:$382631 $
Artin number field: Splitting field of $f= x^{7} - x^{5} + x^{3} - x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_7$
Parity: Odd
Determinant: 1.382631.2t1.1c1

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:
$r_{ 1 }$ $=$ $ 2 a + 70 + \left(35 a + 89\right)\cdot 103 + \left(31 a + 99\right)\cdot 103^{2} + \left(24 a + 3\right)\cdot 103^{3} + \left(88 a + 70\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 55 a + 3 + \left(50 a + 68\right)\cdot 103 + \left(39 a + 54\right)\cdot 103^{2} + \left(27 a + 21\right)\cdot 103^{3} + \left(21 a + 78\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 48 a + 58 + \left(52 a + 63\right)\cdot 103 + \left(63 a + 43\right)\cdot 103^{2} + \left(75 a + 9\right)\cdot 103^{3} + \left(81 a + 72\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 101 a + 72 + \left(67 a + 19\right)\cdot 103 + \left(71 a + 96\right)\cdot 103^{2} + \left(78 a + 99\right)\cdot 103^{3} + \left(14 a + 30\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 19 + 14\cdot 103 + 53\cdot 103^{2} + 69\cdot 103^{3} + 50\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 101 + 77\cdot 103 + 66\cdot 103^{2} + 92\cdot 103^{3} + 70\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 89 + 78\cdot 103 + 100\cdot 103^{2} + 11\cdot 103^{3} + 39\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.