Properties

Label 4.21227e3.10t12.1c1
Dimension 4
Group $S_5$
Conductor $ 21227^{3}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:$9564579024083= 21227^{3} $
Artin number field: Splitting field of $f= x^{5} - x^{4} - 4 x^{2} + 3 x + 2 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_5$
Parity: Odd
Determinant: 1.21227.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$: $ x^{2} + 6 x + 3 $
Roots: \[ \begin{aligned} r_{ 1 } &= 2 a + 1 + \left(a + 6\right)\cdot 7 + \left(5 a + 5\right)\cdot 7^{2} + \left(a + 5\right)\cdot 7^{3} + \left(5 a + 4\right)\cdot 7^{4} +O\left(7^{ 5 }\right) \\ r_{ 2 } &= 6 + 5\cdot 7 + 4\cdot 7^{2} + 7^{3} + 6\cdot 7^{4} +O\left(7^{ 5 }\right) \\ r_{ 3 } &= 6 a + 3 + \left(3 a + 6\right)\cdot 7 + \left(a + 4\right)\cdot 7^{2} + 4 a\cdot 7^{3} + \left(4 a + 4\right)\cdot 7^{4} +O\left(7^{ 5 }\right) \\ r_{ 4 } &= 5 a + 3 + \left(5 a + 5\right)\cdot 7 + \left(a + 2\right)\cdot 7^{2} + \left(5 a + 2\right)\cdot 7^{3} + \left(a + 1\right)\cdot 7^{4} +O\left(7^{ 5 }\right) \\ r_{ 5 } &= a + 2 + \left(3 a + 4\right)\cdot 7 + \left(5 a + 2\right)\cdot 7^{2} + \left(2 a + 3\right)\cdot 7^{3} + \left(2 a + 4\right)\cdot 7^{4} +O\left(7^{ 5 }\right) \\ \end{aligned}\]

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$$1$$()$$4$
$10$$2$$(1,2)$$-2$
$15$$2$$(1,2)(3,4)$$0$
$20$$3$$(1,2,3)$$1$
$30$$4$$(1,2,3,4)$$0$
$24$$5$$(1,2,3,4,5)$$-1$
$20$$6$$(1,2,3)(4,5)$$1$
The blue line marks the conjugacy class containing complex conjugation.