Properties

Label 6.277_811.7t7.1c1
Dimension 6
Group $S_7$
Conductor $ 277 \cdot 811 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$6$
Group:$S_7$
Conductor:$224647= 277 \cdot 811 $
Artin number field: Splitting field of $f= x^{7} - 2 x^{5} - x^{4} + 2 x^{3} + 2 x^{2} - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_7$
Parity: Odd
Determinant: 1.277_811.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 151 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 151 }$: $ x^{2} + 149 x + 6 $
Roots:
$r_{ 1 }$ $=$ $ 142 a + 8 + \left(17 a + 84\right)\cdot 151 + \left(49 a + 63\right)\cdot 151^{2} + \left(61 a + 45\right)\cdot 151^{3} + \left(140 a + 126\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 10 + 138\cdot 151 + 38\cdot 151^{2} + 62\cdot 151^{3} + 91\cdot 151^{4} +O\left(151^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 95 a + 133 + \left(7 a + 97\right)\cdot 151 + \left(14 a + 53\right)\cdot 151^{2} + \left(93 a + 149\right)\cdot 151^{3} + \left(44 a + 107\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 128 a + 93 + \left(143 a + 139\right)\cdot 151 + \left(44 a + 141\right)\cdot 151^{2} + \left(91 a + 110\right)\cdot 151^{3} + \left(35 a + 150\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 56 a + 21 + \left(143 a + 18\right)\cdot 151 + \left(136 a + 74\right)\cdot 151^{2} + \left(57 a + 19\right)\cdot 151^{3} + \left(106 a + 104\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 9 a + 141 + \left(133 a + 128\right)\cdot 151 + \left(101 a + 143\right)\cdot 151^{2} + \left(89 a + 118\right)\cdot 151^{3} + \left(10 a + 43\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 23 a + 47 + \left(7 a + 148\right)\cdot 151 + \left(106 a + 87\right)\cdot 151^{2} + \left(59 a + 97\right)\cdot 151^{3} + \left(115 a + 130\right)\cdot 151^{4} +O\left(151^{ 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.