Properties

Label 5.268...213.6t14.a.a
Dimension $5$
Group $S_5$
Conductor $2.681\times 10^{15}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $5$
Group: $S_5$
Conductor: \(2680810943141213\)\(\medspace = 138917^{3} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 5.5.138917.1
Galois orbit size: $1$
Smallest permutation container: $\PGL(2,5)$
Parity: even
Determinant: 1.138917.2t1.a.a
Projective image: $S_5$
Projective stem field: Galois closure of 5.5.138917.1

Defining polynomial

$f(x)$$=$ \( x^{5} - 6x^{3} - 2x^{2} + 3x + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: \( x^{2} + 60x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 55 a + 5 + \left(38 a + 32\right)\cdot 61 + \left(22 a + 40\right)\cdot 61^{2} + \left(58 a + 48\right)\cdot 61^{3} + \left(54 a + 42\right)\cdot 61^{4} +O(61^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 26 + 29\cdot 61 + 10\cdot 61^{2} + 45\cdot 61^{3} + 3\cdot 61^{4} +O(61^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 34 a + 29 + \left(18 a + 60\right)\cdot 61 + \left(50 a + 37\right)\cdot 61^{2} + \left(56 a + 29\right)\cdot 61^{3} + \left(57 a + 17\right)\cdot 61^{4} +O(61^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 6 a + 60 + \left(22 a + 15\right)\cdot 61 + \left(38 a + 24\right)\cdot 61^{2} + \left(2 a + 23\right)\cdot 61^{3} + \left(6 a + 39\right)\cdot 61^{4} +O(61^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 27 a + 2 + \left(42 a + 45\right)\cdot 61 + \left(10 a + 8\right)\cdot 61^{2} + \left(4 a + 36\right)\cdot 61^{3} + \left(3 a + 18\right)\cdot 61^{4} +O(61^{5})\) 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.