Properties

Label 4.641...287.10t12.a.a
Dimension $4$
Group $S_5$
Conductor $6.417\times 10^{12}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $4$
Group: $S_5$
Conductor: \(6417228161287\)\(\medspace = 18583^{3}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 5.3.18583.1
Galois orbit size: $1$
Smallest permutation container: $S_5$
Parity: odd
Determinant: 1.18583.2t1.a.a
Projective image: $S_5$
Projective stem field: 5.3.18583.1

Defining polynomial

$f(x)$$=$\(x^{5} - x^{4} - x^{3} - 2 x^{2} + x + 1\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$: \(x^{2} + 6 x + 3\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 2 a + 6 + \left(2 a + 1\right)\cdot 7 + \left(2 a + 6\right)\cdot 7^{2} + 4 a\cdot 7^{3} + 5\cdot 7^{4} +O(7^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 2 a + 4 + 4 a\cdot 7 + \left(4 a + 3\right)\cdot 7^{2} + \left(3 a + 1\right)\cdot 7^{3} + \left(4 a + 1\right)\cdot 7^{4} +O(7^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 5 a + 1 + \left(4 a + 2\right)\cdot 7 + \left(4 a + 6\right)\cdot 7^{2} + \left(2 a + 2\right)\cdot 7^{3} + \left(6 a + 1\right)\cdot 7^{4} +O(7^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 5 + 6\cdot 7 + 7^{2} + 7^{3} + 4\cdot 7^{4} +O(7^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 5 a + 6 + \left(2 a + 2\right)\cdot 7 + \left(2 a + 3\right)\cdot 7^{2} + 3 a\cdot 7^{3} + \left(2 a + 2\right)\cdot 7^{4} +O(7^{5})\)  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.