Properties

Label 3.989.6t11.b.a
Dimension $3$
Group $S_4\times C_2$
Conductor $989$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $S_4\times C_2$
Conductor: \(989\)\(\medspace = 23 \cdot 43 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.0.22747.1
Galois orbit size: $1$
Smallest permutation container: $S_4\times C_2$
Parity: even
Determinant: 1.989.2t1.a.a
Projective image: $S_4$
Projective stem field: Galois closure of 4.2.42527.1

Defining polynomial

$f(x)$$=$ \( x^{6} - x^{5} - 2x^{3} + x^{2} + x + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: \( x^{2} + 58x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 28 a + 19 + 33\cdot 59 + \left(43 a + 57\right)\cdot 59^{2} + \left(58 a + 39\right)\cdot 59^{3} + \left(8 a + 25\right)\cdot 59^{4} + \left(30 a + 4\right)\cdot 59^{5} +O(59^{6})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 31 a + 47 + \left(58 a + 5\right)\cdot 59 + \left(15 a + 41\right)\cdot 59^{2} + 55\cdot 59^{3} + \left(50 a + 34\right)\cdot 59^{4} + \left(28 a + 25\right)\cdot 59^{5} +O(59^{6})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 39 a + 32 + \left(17 a + 49\right)\cdot 59 + \left(7 a + 18\right)\cdot 59^{2} + \left(9 a + 49\right)\cdot 59^{3} + \left(56 a + 8\right)\cdot 59^{4} + \left(39 a + 15\right)\cdot 59^{5} +O(59^{6})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 25 + 20\cdot 59^{2} + 4\cdot 59^{3} + 39\cdot 59^{4} + 51\cdot 59^{5} +O(59^{6})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 43 + 31\cdot 59^{2} + 35\cdot 59^{3} + 12\cdot 59^{4} + 22\cdot 59^{5} +O(59^{6})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 20 a + 12 + \left(41 a + 28\right)\cdot 59 + \left(51 a + 8\right)\cdot 59^{2} + \left(49 a + 51\right)\cdot 59^{3} + \left(2 a + 55\right)\cdot 59^{4} + \left(19 a + 57\right)\cdot 59^{5} +O(59^{6})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

Cycle notation
$(1,3,4)(2,6,5)$
$(3,4)(5,6)$
$(3,6)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$3$
$1$$2$$(1,2)(3,6)(4,5)$$-3$
$3$$2$$(1,2)$$1$
$3$$2$$(1,2)(3,6)$$-1$
$6$$2$$(3,4)(5,6)$$1$
$6$$2$$(1,2)(3,4)(5,6)$$-1$
$8$$3$$(1,3,4)(2,6,5)$$0$
$6$$4$$(1,6,2,3)$$1$
$6$$4$$(1,5,2,4)(3,6)$$-1$
$8$$6$$(1,6,5,2,3,4)$$0$

The blue line marks the conjugacy class containing complex conjugation.