Properties

Label 6.19_18413.7t7.1c1
Dimension 6
Group $S_7$
Conductor $ 19 \cdot 18413 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$6$
Group:$S_7$
Conductor:$349847= 19 \cdot 18413 $
Artin number field: Splitting field of $f= x^{7} - x^{6} + x^{5} - x^{4} + x^{2} - x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_7$
Parity: Odd
Determinant: 1.19_18413.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 89 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 89 }$: $ x^{2} + 82 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 35 a + 73 + \left(13 a + 3\right)\cdot 89 + \left(49 a + 16\right)\cdot 89^{2} + \left(2 a + 43\right)\cdot 89^{3} + \left(16 a + 58\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 67 + 28\cdot 89 + 19\cdot 89^{2} + 84\cdot 89^{3} + 19\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 53 a + 10 + \left(41 a + 26\right)\cdot 89 + \left(2 a + 81\right)\cdot 89^{2} + \left(51 a + 20\right)\cdot 89^{3} + \left(2 a + 47\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 36 a + 25 + \left(47 a + 86\right)\cdot 89 + \left(86 a + 56\right)\cdot 89^{2} + \left(37 a + 19\right)\cdot 89^{3} + \left(86 a + 14\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 54 a + 51 + \left(75 a + 62\right)\cdot 89 + \left(39 a + 79\right)\cdot 89^{2} + \left(86 a + 11\right)\cdot 89^{3} + \left(72 a + 79\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 71 a + 84 + \left(71 a + 80\right)\cdot 89 + \left(31 a + 64\right)\cdot 89^{2} + \left(70 a + 35\right)\cdot 89^{3} + \left(31 a + 81\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 18 a + 47 + \left(17 a + 67\right)\cdot 89 + \left(57 a + 37\right)\cdot 89^{2} + \left(18 a + 51\right)\cdot 89^{3} + \left(57 a + 55\right)\cdot 89^{4} +O\left(89^{ 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.