Properties

Label 6.5611284433.20t30.a.a
Dimension $6$
Group $S_5$
Conductor $5611284433$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $6$
Group: $S_5$
Conductor: \(5611284433\)\(\medspace = 1777^{3}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 5.1.1777.1
Galois orbit size: $1$
Smallest permutation container: 20T30
Parity: even
Determinant: 1.1777.2t1.a.a
Projective image: $S_5$
Projective stem field: 5.1.1777.1

Defining polynomial

$f(x)$$=$\(x^{5} - x^{4} + x^{3} - 2 x^{2} + x - 1\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: \(x^{2} + 21 x + 5\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 3 a + 8 + \left(11 a + 8\right)\cdot 23 + \left(17 a + 16\right)\cdot 23^{2} + \left(13 a + 3\right)\cdot 23^{3} + \left(17 a + 13\right)\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 20 a + 14 + \left(11 a + 4\right)\cdot 23 + \left(5 a + 17\right)\cdot 23^{2} + \left(9 a + 13\right)\cdot 23^{3} + \left(5 a + 11\right)\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 2 a + 3 + \left(16 a + 17\right)\cdot 23 + \left(12 a + 1\right)\cdot 23^{2} + \left(8 a + 3\right)\cdot 23^{3} + \left(18 a + 7\right)\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 15 + 14\cdot 23 + 22\cdot 23^{2} + 17\cdot 23^{3} + 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 21 a + 7 + \left(6 a + 1\right)\cdot 23 + \left(10 a + 11\right)\cdot 23^{2} + \left(14 a + 7\right)\cdot 23^{3} + \left(4 a + 12\right)\cdot 23^{4} +O(23^{5})\)  Toggle raw display

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.