Properties

Label 5.167_313.6t16.1c1
Dimension 5
Group $S_6$
Conductor $ 167 \cdot 313 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:$52271= 167 \cdot 313 $
Artin number field: Splitting field of $f= x^{6} - x^{5} + x^{4} - x^{3} + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_6$
Parity: Odd
Determinant: 1.167_313.2t1.1c1

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:
$r_{ 1 }$ $=$ $ 33 a + 18 + \left(56 a + 13\right)\cdot 83 + \left(57 a + 65\right)\cdot 83^{2} + \left(63 a + 10\right)\cdot 83^{3} + \left(26 a + 17\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 37 + 68\cdot 83 + 49\cdot 83^{2} + 29\cdot 83^{3} + 70\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 50 a + 51 + \left(26 a + 36\right)\cdot 83 + \left(25 a + 66\right)\cdot 83^{2} + \left(19 a + 16\right)\cdot 83^{3} + \left(56 a + 63\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 51 + 76\cdot 83 + 17\cdot 83^{2} + 52\cdot 83^{3} + 54\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 50 + 67\cdot 83 + 49\cdot 83^{2} + 35\cdot 83^{3} + 18\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 43 + 69\cdot 83 + 82\cdot 83^{2} + 20\cdot 83^{3} + 25\cdot 83^{4} +O\left(83^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

Cycle notation
$(1,2)$
$(1,2,3,4,5,6)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$5$
$15$$2$$(1,2)(3,4)(5,6)$$-1$
$15$$2$$(1,2)$$3$
$45$$2$$(1,2)(3,4)$$1$
$40$$3$$(1,2,3)(4,5,6)$$-1$
$40$$3$$(1,2,3)$$2$
$90$$4$$(1,2,3,4)(5,6)$$-1$
$90$$4$$(1,2,3,4)$$1$
$144$$5$$(1,2,3,4,5)$$0$
$120$$6$$(1,2,3,4,5,6)$$-1$
$120$$6$$(1,2,3)(4,5)$$0$
The blue line marks the conjugacy class containing complex conjugation.