Properties

Label 4.3089.5t5.b.a
Dimension $4$
Group $S_5$
Conductor $3089$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $4$
Group: $S_5$
Conductor: \(3089\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 5.1.3089.1
Galois orbit size: $1$
Smallest permutation container: $S_5$
Parity: even
Determinant: 1.3089.2t1.a.a
Projective image: $S_5$
Projective stem field: 5.1.3089.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \(x^{2} + 16 x + 3\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 3 + 12\cdot 17 + 17^{2} + 16\cdot 17^{3} + 17^{4} +O(17^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 13 + 3\cdot 17 + 2\cdot 17^{2} + 8\cdot 17^{3} + 2\cdot 17^{4} +O(17^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 5 + 3\cdot 17 + 13\cdot 17^{2} + 14\cdot 17^{3} + 6\cdot 17^{4} +O(17^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 15 a + 16 + \left(11 a + 8\right)\cdot 17 + \left(8 a + 1\right)\cdot 17^{2} + \left(6 a + 7\right)\cdot 17^{3} + \left(3 a + 4\right)\cdot 17^{4} +O(17^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 2 a + 14 + \left(5 a + 5\right)\cdot 17 + \left(8 a + 15\right)\cdot 17^{2} + \left(10 a + 4\right)\cdot 17^{3} + \left(13 a + 1\right)\cdot 17^{4} +O(17^{5})\)  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.