Properties

Label 5.180649580077.6t14.a.a
Dimension $5$
Group $S_5$
Conductor $180649580077$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $5$
Group: $S_5$
Conductor: \(180649580077\)\(\medspace = 5653^{3} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 5.1.5653.1
Galois orbit size: $1$
Smallest permutation container: $\PGL(2,5)$
Parity: even
Determinant: 1.5653.2t1.a.a
Projective image: $S_5$
Projective stem field: Galois closure of 5.1.5653.1

Defining polynomial

$f(x)$$=$ \( x^{5} - x^{4} + 2x - 1 \) Copy content 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 $\mathbb{Q}_{ 23 }$: \( x^{2} + 21x + 5 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 21 a + 20 + \left(10 a + 10\right)\cdot 23 + \left(4 a + 15\right)\cdot 23^{2} + \left(18 a + 17\right)\cdot 23^{3} + 8 a\cdot 23^{4} +O(23^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 17 + 2\cdot 23 + 20\cdot 23^{2} + 17\cdot 23^{3} + 21\cdot 23^{4} +O(23^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 2 a + 16 + \left(12 a + 11\right)\cdot 23 + \left(18 a + 13\right)\cdot 23^{2} + \left(4 a + 3\right)\cdot 23^{3} + 14 a\cdot 23^{4} +O(23^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 11 a + 9 + \left(21 a + 17\right)\cdot 23 + \left(5 a + 14\right)\cdot 23^{2} + \left(19 a + 21\right)\cdot 23^{3} + \left(17 a + 14\right)\cdot 23^{4} +O(23^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 12 a + 8 + \left(a + 3\right)\cdot 23 + \left(17 a + 5\right)\cdot 23^{2} + \left(3 a + 8\right)\cdot 23^{3} + \left(5 a + 8\right)\cdot 23^{4} +O(23^{5})\) Copy content 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$$1$$()$$5$
$10$$2$$(1,2)$$-1$
$15$$2$$(1,2)(3,4)$$1$
$20$$3$$(1,2,3)$$-1$
$30$$4$$(1,2,3,4)$$1$
$24$$5$$(1,2,3,4,5)$$0$
$20$$6$$(1,2,3)(4,5)$$-1$

The blue line marks the conjugacy class containing complex conjugation.