Properties

Label 2.1444.6t5.b.b
Dimension $2$
Group $S_3\times C_3$
Conductor $1444$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $S_3\times C_3$
Conductor: \(1444\)\(\medspace = 2^{2} \cdot 19^{2}\)
Artin stem field: 6.0.39617584.1
Galois orbit size: $2$
Smallest permutation container: $S_3\times C_3$
Parity: odd
Determinant: 1.19.6t1.a.a
Projective image: $S_3$
Projective stem field: 3.1.76.1

Defining polynomial

$f(x)$$=$\(x^{6} - x^{5} + 2 x^{4} - 11 x^{3} + 18 x^{2} + 33 x + 83\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: \(x^{2} + 29 x + 3\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 28 a + 3 + \left(2 a + 8\right)\cdot 31 + \left(10 a + 20\right)\cdot 31^{2} + \left(29 a + 13\right)\cdot 31^{3} + 3 a\cdot 31^{4} + \left(12 a + 14\right)\cdot 31^{5} +O(31^{6})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 4 a + 6 + \left(2 a + 18\right)\cdot 31 + \left(14 a + 2\right)\cdot 31^{2} + \left(5 a + 21\right)\cdot 31^{3} + \left(26 a + 2\right)\cdot 31^{4} + \left(a + 12\right)\cdot 31^{5} +O(31^{6})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 3 a + 28 + \left(28 a + 16\right)\cdot 31 + \left(20 a + 6\right)\cdot 31^{2} + a\cdot 31^{3} + \left(27 a + 10\right)\cdot 31^{4} + \left(18 a + 3\right)\cdot 31^{5} +O(31^{6})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 6 a + \left(12 a + 22\right)\cdot 31 + \left(16 a + 22\right)\cdot 31^{2} + \left(5 a + 22\right)\cdot 31^{3} + \left(4 a + 13\right)\cdot 31^{4} + \left(18 a + 5\right)\cdot 31^{5} +O(31^{6})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 27 a + 14 + \left(28 a + 18\right)\cdot 31 + \left(16 a + 28\right)\cdot 31^{2} + \left(25 a + 17\right)\cdot 31^{3} + \left(4 a + 18\right)\cdot 31^{4} + \left(29 a + 20\right)\cdot 31^{5} +O(31^{6})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 25 a + 12 + \left(18 a + 9\right)\cdot 31 + \left(14 a + 12\right)\cdot 31^{2} + \left(25 a + 17\right)\cdot 31^{3} + \left(26 a + 16\right)\cdot 31^{4} + \left(12 a + 6\right)\cdot 31^{5} +O(31^{6})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.