Properties

Label 6.48615135735473.20t30.a.a
Dimension 6
Group $S_5$
Conductor $ 36497^{3}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$6$
Group:$S_5$
Conductor:$48615135735473= 36497^{3} $
Artin number field: Splitting field of 5.5.36497.1 defined by $f= x^{5} - 2 x^{4} - 3 x^{3} + 5 x^{2} + x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: 20T30
Parity: Even
Determinant: 1.36497.2t1.a.a
Projective image: $S_5$
Projective field: Galois closure of 5.5.36497.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $ x^{2} + 29 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 5 a + 3 + \left(4 a + 27\right)\cdot 31 + \left(11 a + 28\right)\cdot 31^{2} + \left(19 a + 28\right)\cdot 31^{3} + \left(11 a + 8\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 7 a + 15 + \left(18 a + 24\right)\cdot 31 + \left(12 a + 2\right)\cdot 31^{2} + \left(6 a + 1\right)\cdot 31^{3} + \left(28 a + 18\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 26 a + 13 + \left(26 a + 30\right)\cdot 31 + \left(19 a + 15\right)\cdot 31^{2} + \left(11 a + 25\right)\cdot 31^{3} + \left(19 a + 12\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 4 + 19\cdot 31 + 4\cdot 31^{2} + 5\cdot 31^{3} + 16\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 24 a + 29 + \left(12 a + 22\right)\cdot 31 + \left(18 a + 9\right)\cdot 31^{2} + \left(24 a + 1\right)\cdot 31^{3} + \left(2 a + 6\right)\cdot 31^{4} +O\left(31^{ 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.