Properties

Label 6.240881e3.20t35.1c1
Dimension 6
Group $S_5$
Conductor $ 240881^{3}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$6$
Group:$S_5$
Conductor:$13976796319717841= 240881^{3} $
Artin number field: Splitting field of $f= x^{5} - 2 x^{4} - 5 x^{3} + 9 x^{2} + 5 x - 7 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: 20T35
Parity: Even
Determinant: 1.240881.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{2} + 24 x + 2 $
Roots: \[ \begin{aligned} r_{ 1 } &= 18 a + 6 + \left(2 a + 21\right)\cdot 29 + 8\cdot 29^{2} + \left(a + 27\right)\cdot 29^{3} + \left(25 a + 6\right)\cdot 29^{4} +O\left(29^{ 5 }\right) \\ r_{ 2 } &= 7 + 12\cdot 29 + 13\cdot 29^{2} + 4\cdot 29^{3} + 11\cdot 29^{4} +O\left(29^{ 5 }\right) \\ r_{ 3 } &= 6 a + 4 + \left(11 a + 8\right)\cdot 29 + \left(8 a + 28\right)\cdot 29^{2} + \left(28 a + 2\right)\cdot 29^{3} + \left(21 a + 15\right)\cdot 29^{4} +O\left(29^{ 5 }\right) \\ r_{ 4 } &= 23 a + 5 + 17 a\cdot 29 + \left(20 a + 1\right)\cdot 29^{2} + 20\cdot 29^{3} + \left(7 a + 9\right)\cdot 29^{4} +O\left(29^{ 5 }\right) \\ r_{ 5 } &= 11 a + 9 + \left(26 a + 16\right)\cdot 29 + \left(28 a + 6\right)\cdot 29^{2} + \left(27 a + 3\right)\cdot 29^{3} + \left(3 a + 15\right)\cdot 29^{4} +O\left(29^{ 5 }\right) \\ \end{aligned}\]

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.