Properties

Label 6.163909850319.20t30.a.a
Dimension 6
Group $S_5$
Conductor $ 3^{5} \cdot 877^{3}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$6$
Group:$S_5$
Conductor:$163909850319= 3^{5} \cdot 877^{3} $
Artin number field: Splitting field of 5.3.23679.1 defined by $f= x^{5} - 2 x^{4} + x^{2} - 2 x + 3 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: 20T30
Parity: Odd
Determinant: 1.2631.2t1.a.a
Projective image: $S_5$
Projective field: Galois closure of 5.3.23679.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{2} + 45 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 14 a + 14 + \left(13 a + 46\right)\cdot 47 + \left(9 a + 41\right)\cdot 47^{2} + \left(31 a + 30\right)\cdot 47^{3} + \left(5 a + 33\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 33 a + 42 + \left(33 a + 11\right)\cdot 47 + 37 a\cdot 47^{2} + \left(15 a + 37\right)\cdot 47^{3} + \left(41 a + 13\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 39 a + 11 + \left(22 a + 27\right)\cdot 47 + \left(45 a + 17\right)\cdot 47^{2} + \left(17 a + 7\right)\cdot 47^{3} + \left(24 a + 38\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 8 a + 42 + \left(24 a + 33\right)\cdot 47 + \left(a + 38\right)\cdot 47^{2} + \left(29 a + 44\right)\cdot 47^{3} + \left(22 a + 21\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 34 + 21\cdot 47 + 42\cdot 47^{2} + 20\cdot 47^{3} + 33\cdot 47^{4} +O\left(47^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 5 }$

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 5 }$ Character value
$1$$1$$()$$6$
$10$$2$$(1,2)$$0$
$15$$2$$(1,2)(3,4)$$-2$
$20$$3$$(1,2,3)$$0$
$30$$4$$(1,2,3,4)$$0$
$24$$5$$(1,2,3,4,5)$$1$
$20$$6$$(1,2,3)(4,5)$$0$
The blue line marks the conjugacy class containing complex conjugation.