Properties

Label 6.83_3037.7t7.1c1
Dimension 6
Group $S_7$
Conductor $ 83 \cdot 3037 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$6$
Group:$S_7$
Conductor:$252071= 83 \cdot 3037 $
Artin number field: Splitting field of $f= x^{7} - x^{6} + 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: 1.83_3037.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 167 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 167 }$: $ x^{2} + 166 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 144 + 16\cdot 167 + 146\cdot 167^{2} + 130\cdot 167^{3} + 61\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 58 + 84\cdot 167 + 31\cdot 167^{2} + 68\cdot 167^{3} + 89\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 83 a + 16 + \left(72 a + 80\right)\cdot 167 + \left(71 a + 93\right)\cdot 167^{2} + \left(14 a + 104\right)\cdot 167^{3} + \left(48 a + 10\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 35 a + 41 + \left(126 a + 134\right)\cdot 167 + \left(62 a + 7\right)\cdot 167^{2} + \left(129 a + 10\right)\cdot 167^{3} + \left(41 a + 148\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 84 a + 99 + \left(94 a + 69\right)\cdot 167 + \left(95 a + 92\right)\cdot 167^{2} + \left(152 a + 47\right)\cdot 167^{3} + \left(118 a + 44\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 132 a + 76 + \left(40 a + 58\right)\cdot 167 + \left(104 a + 111\right)\cdot 167^{2} + \left(37 a + 76\right)\cdot 167^{3} + \left(125 a + 60\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 68 + 57\cdot 167 + 18\cdot 167^{2} + 63\cdot 167^{3} + 86\cdot 167^{4} +O\left(167^{ 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.