Properties

Label 6.7_48313.7t7.1c1
Dimension 6
Group $S_7$
Conductor $ 7 \cdot 48313 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$6$
Group:$S_7$
Conductor:$338191= 7 \cdot 48313 $
Artin number field: Splitting field of $f= x^{7} - x^{4} - 2 x^{3} + 2 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_7$
Parity: Odd
Determinant: 1.7_48313.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 113 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 113 }$: $ x^{2} + 101 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 92 a + 87 + \left(91 a + 82\right)\cdot 113 + \left(21 a + 110\right)\cdot 113^{2} + \left(101 a + 14\right)\cdot 113^{3} + \left(26 a + 34\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 21 a + 61 + \left(21 a + 75\right)\cdot 113 + \left(91 a + 54\right)\cdot 113^{2} + \left(11 a + 77\right)\cdot 113^{3} + \left(86 a + 29\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 76 + 13\cdot 113 + 46\cdot 113^{2} + 81\cdot 113^{3} + 68\cdot 113^{4} +O\left(113^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 24 a + 12 + \left(5 a + 33\right)\cdot 113 + \left(22 a + 41\right)\cdot 113^{2} + \left(3 a + 16\right)\cdot 113^{3} + \left(9 a + 52\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 72 + 90\cdot 113 + 77\cdot 113^{2} + 12\cdot 113^{3} +O\left(113^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 70 + 84\cdot 113 + 46\cdot 113^{2} + 103\cdot 113^{3} + 109\cdot 113^{4} +O\left(113^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 89 a + 74 + \left(107 a + 71\right)\cdot 113 + \left(90 a + 74\right)\cdot 113^{2} + \left(109 a + 32\right)\cdot 113^{3} + \left(103 a + 44\right)\cdot 113^{4} +O\left(113^{ 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.