Properties

Label 6.199559.7t7.1c1
Dimension 6
Group $S_7$
Conductor $ 199559 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$6$
Group:$S_7$
Conductor:$199559 $
Artin number field: Splitting field of $f= x^{7} - x^{6} + x^{3} - x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_7$
Parity: Odd
Determinant: 1.199559.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 193 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 193 }$: $ x^{2} + 192 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 23 a + 18 + \left(190 a + 60\right)\cdot 193 + \left(181 a + 46\right)\cdot 193^{2} + \left(105 a + 42\right)\cdot 193^{3} + \left(13 a + 160\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 57 a + 120 + \left(112 a + 18\right)\cdot 193 + \left(104 a + 55\right)\cdot 193^{2} + \left(125 a + 189\right)\cdot 193^{3} + \left(90 a + 151\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 177 + 7\cdot 193 + 84\cdot 193^{2} + 133\cdot 193^{3} + 169\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 169 + 183\cdot 193 + 61\cdot 193^{2} + 130\cdot 193^{3} +O\left(193^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 170 a + 41 + \left(2 a + 34\right)\cdot 193 + \left(11 a + 38\right)\cdot 193^{2} + \left(87 a + 159\right)\cdot 193^{3} + \left(179 a + 67\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 71 + 7\cdot 193 + 53\cdot 193^{2} + 100\cdot 193^{3} + 104\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 136 a + 177 + \left(80 a + 73\right)\cdot 193 + \left(88 a + 47\right)\cdot 193^{2} + \left(67 a + 17\right)\cdot 193^{3} + \left(102 a + 117\right)\cdot 193^{4} +O\left(193^{ 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.