Properties

Label 2.455.6t5.a.a
Dimension $2$
Group $S_3\times C_3$
Conductor $455$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $S_3\times C_3$
Conductor: \(455\)\(\medspace = 5 \cdot 7 \cdot 13 \)
Artin stem field: 6.0.7245875.1
Galois orbit size: $2$
Smallest permutation container: $S_3\times C_3$
Parity: odd
Determinant: 1.455.6t1.b.a
Projective image: $S_3$
Projective stem field: 3.1.5915.1

Defining polynomial

$f(x)$$=$\(x^{6} - x^{5} + 2 x^{4} - 13 x^{3} + 3 x^{2} + 14 x + 9\)  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:
$r_{ 1 }$ $=$ \( a + \left(17 a + 7\right)\cdot 31 + \left(12 a + 21\right)\cdot 31^{2} + \left(17 a + 6\right)\cdot 31^{3} + \left(7 a + 16\right)\cdot 31^{4} + \left(25 a + 30\right)\cdot 31^{5} +O(31^{6})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 30 a + 2 + \left(13 a + 9\right)\cdot 31 + \left(18 a + 29\right)\cdot 31^{2} + \left(13 a + 28\right)\cdot 31^{3} + \left(23 a + 13\right)\cdot 31^{4} + \left(5 a + 11\right)\cdot 31^{5} +O(31^{6})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 7 a + 27 + \left(20 a + 26\right)\cdot 31 + \left(14 a + 25\right)\cdot 31^{2} + \left(15 a + 15\right)\cdot 31^{3} + \left(14 a + 3\right)\cdot 31^{4} + \left(5 a + 19\right)\cdot 31^{5} +O(31^{6})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 24 a + 10 + \left(10 a + 29\right)\cdot 31 + \left(16 a + 3\right)\cdot 31^{2} + \left(15 a + 1\right)\cdot 31^{3} + \left(16 a + 17\right)\cdot 31^{4} + \left(25 a + 15\right)\cdot 31^{5} +O(31^{6})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 12 a + \left(19 a + 28\right)\cdot 31 + \left(20 a + 10\right)\cdot 31^{2} + \left(22 a + 23\right)\cdot 31^{3} + \left(16 a + 15\right)\cdot 31^{4} + \left(10 a + 21\right)\cdot 31^{5} +O(31^{6})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 19 a + 24 + \left(11 a + 23\right)\cdot 31 + \left(10 a + 1\right)\cdot 31^{2} + \left(8 a + 17\right)\cdot 31^{3} + \left(14 a + 26\right)\cdot 31^{4} + \left(20 a + 25\right)\cdot 31^{5} +O(31^{6})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.