Properties

Label 5.31223.6t16.1c1
Dimension 5
Group $S_6$
Conductor $ 31223 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:$31223 $
Artin number field: Splitting field of $f= x^{6} - x^{5} + x^{4} - x^{3} + 2 x^{2} - x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_6$
Parity: Odd
Determinant: 1.31223.2t1.1c1

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:
$r_{ 1 }$ $=$ $ 55 a + 45 + \left(70 a + 11\right)\cdot 89 + \left(12 a + 67\right)\cdot 89^{2} + \left(60 a + 88\right)\cdot 89^{3} + \left(67 a + 53\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 34 a + 74 + \left(18 a + 5\right)\cdot 89 + \left(76 a + 86\right)\cdot 89^{2} + \left(28 a + 51\right)\cdot 89^{3} + \left(21 a + 22\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 54 + 37\cdot 89 + 42\cdot 89^{3} + 23\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 42 a + 32 + \left(33 a + 14\right)\cdot 89 + \left(57 a + 85\right)\cdot 89^{2} + \left(57 a + 87\right)\cdot 89^{3} + \left(68 a + 72\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 4 + 80\cdot 89 + 19\cdot 89^{2} + 7\cdot 89^{3} + 43\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 47 a + 59 + \left(55 a + 28\right)\cdot 89 + \left(31 a + 8\right)\cdot 89^{2} + \left(31 a + 78\right)\cdot 89^{3} + \left(20 a + 50\right)\cdot 89^{4} +O\left(89^{ 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.