Properties

Label 6.389_967.7t7.1c1
Dimension 6
Group $S_7$
Conductor $ 389 \cdot 967 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$6$
Group:$S_7$
Conductor:$376163= 389 \cdot 967 $
Artin number field: Splitting field of $f= x^{7} - x^{6} - x^{4} - x^{3} + x^{2} + x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_7$
Parity: Odd
Determinant: 1.389_967.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$: $ x^{2} + 97 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 21 a + 39 + \left(69 a + 61\right)\cdot 101 + \left(36 a + 2\right)\cdot 101^{2} + \left(35 a + 48\right)\cdot 101^{3} + \left(13 a + 73\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 87 a + 70 + \left(91 a + 9\right)\cdot 101 + \left(46 a + 40\right)\cdot 101^{2} + \left(76 a + 51\right)\cdot 101^{3} + \left(67 a + 84\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 50 + 33\cdot 101 + 58\cdot 101^{2} + 33\cdot 101^{3} + 70\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 80 a + 22 + \left(31 a + 14\right)\cdot 101 + \left(64 a + 80\right)\cdot 101^{2} + \left(65 a + 51\right)\cdot 101^{3} + \left(87 a + 91\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 42 a + 21 + \left(70 a + 29\right)\cdot 101 + \left(88 a + 2\right)\cdot 101^{2} + \left(92 a + 15\right)\cdot 101^{3} + \left(43 a + 63\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 14 a + 14 + \left(9 a + 88\right)\cdot 101 + \left(54 a + 34\right)\cdot 101^{2} + \left(24 a + 7\right)\cdot 101^{3} + \left(33 a + 77\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 59 a + 88 + \left(30 a + 66\right)\cdot 101 + \left(12 a + 84\right)\cdot 101^{2} + \left(8 a + 95\right)\cdot 101^{3} + \left(57 a + 44\right)\cdot 101^{4} +O\left(101^{ 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.