Properties

Label 4.283...393.10t12.a.a
Dimension $4$
Group $S_5$
Conductor $2.830\times 10^{14}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $4$
Group: $S_5$
Conductor: \(283036930148393\)\(\medspace = 65657^{3} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 5.5.65657.1
Galois orbit size: $1$
Smallest permutation container: $S_5$
Parity: even
Determinant: 1.65657.2t1.a.a
Projective image: $S_5$
Projective stem field: Galois closure of 5.5.65657.1

Defining polynomial

$f(x)$$=$ \( x^{5} - x^{4} - 5x^{3} + 2x^{2} + 5x + 1 \) 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:
$r_{ 1 }$ $=$ \( 28 a + \left(6 a + 22\right)\cdot 29 + \left(9 a + 14\right)\cdot 29^{2} + \left(21 a + 27\right)\cdot 29^{3} + \left(28 a + 21\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( a + 24 + \left(22 a + 28\right)\cdot 29 + \left(19 a + 24\right)\cdot 29^{2} + \left(7 a + 8\right)\cdot 29^{3} + 28\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 6 a + 6 + \left(14 a + 1\right)\cdot 29 + \left(26 a + 13\right)\cdot 29^{2} + \left(2 a + 10\right)\cdot 29^{3} + \left(4 a + 27\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 22 + 26\cdot 29 + 18\cdot 29^{2} + 12\cdot 29^{3} + 22\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 23 a + 7 + \left(14 a + 8\right)\cdot 29 + \left(2 a + 15\right)\cdot 29^{2} + \left(26 a + 27\right)\cdot 29^{3} + \left(24 a + 15\right)\cdot 29^{4} +O(29^{5})\) 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$$1$$()$$4$
$10$$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.