Properties

Label 6.193327.7t7.1c1
Dimension 6
Group $S_7$
Conductor $ 193327 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$6$
Group:$S_7$
Conductor:$193327 $
Artin number field: Splitting field of $f= x^{7} - x^{6} + 2 x^{4} - 2 x^{3} + 2 x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_7$
Parity: Odd
Determinant: 1.193327.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 83 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 83 }$: $ x^{2} + 82 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 2 + 22\cdot 83 + 22\cdot 83^{2} + 70\cdot 83^{3} + 8\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 32 + 54\cdot 83 + 66\cdot 83^{2} + 19\cdot 83^{3} + 75\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 66 + 10\cdot 83^{2} + 59\cdot 83^{3} + 7\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 16 a + 37 + \left(78 a + 41\right)\cdot 83 + \left(47 a + 58\right)\cdot 83^{2} + \left(70 a + 28\right)\cdot 83^{3} + \left(58 a + 81\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 82 a + 72 + \left(43 a + 73\right)\cdot 83 + \left(14 a + 4\right)\cdot 83^{2} + \left(25 a + 46\right)\cdot 83^{3} + \left(19 a + 47\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 6 }$ $=$ $ a + 71 + \left(39 a + 35\right)\cdot 83 + \left(68 a + 58\right)\cdot 83^{2} + \left(57 a + 56\right)\cdot 83^{3} + \left(63 a + 41\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 67 a + 53 + \left(4 a + 20\right)\cdot 83 + \left(35 a + 28\right)\cdot 83^{2} + \left(12 a + 51\right)\cdot 83^{3} + \left(24 a + 69\right)\cdot 83^{4} +O\left(83^{ 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.