Properties

Label 6.242147.7t7.1c1
Dimension 6
Group $S_7$
Conductor $ 242147 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$6$
Group:$S_7$
Conductor:$242147 $
Artin number field: Splitting field of $f= x^{7} - 2 x^{6} + 2 x^{5} - 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: 1.242147.2t1.1c1

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:
$r_{ 1 }$ $=$ $ 28 + 20\cdot 89 + 50\cdot 89^{2} + 13\cdot 89^{3} + 64\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 67 a + 27 + \left(5 a + 4\right)\cdot 89 + \left(52 a + 6\right)\cdot 89^{2} + \left(38 a + 84\right)\cdot 89^{3} + \left(32 a + 60\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 82 a + \left(41 a + 25\right)\cdot 89 + \left(23 a + 55\right)\cdot 89^{2} + \left(26 a + 58\right)\cdot 89^{3} + \left(23 a + 22\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 7 a + 40 + \left(47 a + 58\right)\cdot 89 + \left(65 a + 88\right)\cdot 89^{2} + \left(62 a + 40\right)\cdot 89^{3} + \left(65 a + 70\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 22 a + 51 + \left(83 a + 66\right)\cdot 89 + \left(36 a + 8\right)\cdot 89^{2} + \left(50 a + 35\right)\cdot 89^{3} + \left(56 a + 71\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 29 + 13\cdot 89 + 18\cdot 89^{2} + 11\cdot 89^{3} + 83\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 5 + 79\cdot 89 + 39\cdot 89^{2} + 23\cdot 89^{3} + 72\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.