Properties

Label 4.142...313.10t12.a.a
Dimension $4$
Group $S_5$
Conductor $1.420\times 10^{13}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $4$
Group: $S_5$
Conductor: \(14202376626313\)\(\medspace = 61^{3} \cdot 397^{3} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 5.5.24217.1
Galois orbit size: $1$
Smallest permutation container: $S_5$
Parity: even
Determinant: 1.24217.2t1.a.a
Projective image: $S_5$
Projective stem field: Galois closure of 5.5.24217.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: \( x^{2} + 45x + 5 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 5 + 19\cdot 47 + 14\cdot 47^{2} + 31\cdot 47^{3} + 6\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 22 a + 6 + \left(9 a + 26\right)\cdot 47 + \left(3 a + 10\right)\cdot 47^{2} + \left(21 a + 28\right)\cdot 47^{3} + \left(31 a + 18\right)\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 25 a + 15 + \left(13 a + 35\right)\cdot 47 + \left(46 a + 14\right)\cdot 47^{2} + \left(24 a + 5\right)\cdot 47^{3} + \left(43 a + 20\right)\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 22 a + 18 + \left(33 a + 37\right)\cdot 47 + 46\cdot 47^{2} + \left(22 a + 8\right)\cdot 47^{3} + \left(3 a + 35\right)\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 25 a + 3 + \left(37 a + 23\right)\cdot 47 + \left(43 a + 7\right)\cdot 47^{2} + \left(25 a + 20\right)\cdot 47^{3} + \left(15 a + 13\right)\cdot 47^{4} +O(47^{5})\) 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.