Properties

Label 6.286711.7t7.1c1
Dimension 6
Group $S_7$
Conductor $ 286711 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$6$
Group:$S_7$
Conductor:$286711 $
Artin number field: Splitting field of $f= x^{7} - x^{5} - 2 x^{4} + x^{3} + 2 x^{2} - x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_7$
Parity: Odd
Determinant: 1.286711.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 191 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 191 }$: $ x^{2} + 190 x + 19 $
Roots:
$r_{ 1 }$ $=$ $ 98 a + 183 + \left(61 a + 66\right)\cdot 191 + \left(127 a + 123\right)\cdot 191^{2} + \left(49 a + 32\right)\cdot 191^{3} + \left(44 a + 122\right)\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 183 a + \left(25 a + 87\right)\cdot 191 + \left(105 a + 67\right)\cdot 191^{2} + \left(20 a + 165\right)\cdot 191^{3} + \left(101 a + 38\right)\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 8 a + 183 + \left(165 a + 120\right)\cdot 191 + \left(85 a + 146\right)\cdot 191^{2} + \left(170 a + 80\right)\cdot 191^{3} + \left(89 a + 119\right)\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 188 a + 131 + \left(108 a + 105\right)\cdot 191 + \left(181 a + 2\right)\cdot 191^{2} + \left(72 a + 37\right)\cdot 191^{3} + \left(152 a + 115\right)\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 3 a + 128 + \left(82 a + 26\right)\cdot 191 + \left(9 a + 75\right)\cdot 191^{2} + \left(118 a + 119\right)\cdot 191^{3} + \left(38 a + 3\right)\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 49 + 135\cdot 191 + 159\cdot 191^{2} + 182\cdot 191^{3} + 56\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 93 a + 90 + \left(129 a + 30\right)\cdot 191 + \left(63 a + 189\right)\cdot 191^{2} + \left(141 a + 145\right)\cdot 191^{3} + \left(146 a + 116\right)\cdot 191^{4} +O\left(191^{ 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.