Properties

Label 6.224...304.20t30.a.a
Dimension $6$
Group $S_5$
Conductor $2.241\times 10^{13}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $6$
Group: $S_5$
Conductor: \(22412680708304\)\(\medspace = 2^{4} \cdot 67^{3} \cdot 167^{3} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 5.5.179024.1
Galois orbit size: $1$
Smallest permutation container: 20T30
Parity: even
Determinant: 1.11189.2t1.a.a
Projective image: $S_5$
Projective stem field: Galois closure of 5.5.179024.1

Defining polynomial

$f(x)$$=$ \( x^{5} - 8x^{3} + 6x - 2 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: \( x^{2} + 24x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 23 a + 23 + \left(20 a + 13\right)\cdot 29 + \left(5 a + 1\right)\cdot 29^{2} + \left(14 a + 9\right)\cdot 29^{3} + \left(16 a + 2\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 15 a + 17 + \left(21 a + 4\right)\cdot 29 + \left(25 a + 2\right)\cdot 29^{2} + \left(14 a + 22\right)\cdot 29^{3} + \left(21 a + 11\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 6 a + 22 + \left(8 a + 7\right)\cdot 29 + \left(23 a + 9\right)\cdot 29^{2} + \left(14 a + 16\right)\cdot 29^{3} + \left(12 a + 12\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 14 a + 5 + \left(7 a + 10\right)\cdot 29 + \left(3 a + 22\right)\cdot 29^{2} + \left(14 a + 12\right)\cdot 29^{3} + \left(7 a + 17\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 20 + 21\cdot 29 + 22\cdot 29^{2} + 26\cdot 29^{3} + 13\cdot 29^{4} +O(29^{5})\) 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$$1$$()$$6$
$10$$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.