Properties

Label 2.140.6t5.a.a
Dimension $2$
Group $S_3\times C_3$
Conductor $140$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $S_3\times C_3$
Conductor: \(140\)\(\medspace = 2^{2} \cdot 5 \cdot 7 \)
Artin stem field: 6.0.392000.1
Galois orbit size: $2$
Smallest permutation container: $S_3\times C_3$
Parity: odd
Determinant: 1.140.6t1.b.a
Projective image: $S_3$
Projective stem field: 3.1.980.1

Defining polynomial

$f(x)$$=$\(x^{6} - 2 x^{5} + 3 x^{4} + 4 x^{2} + 2 x + 1\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 7.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \(x^{2} + 12 x + 2\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 12 a + 6 + \left(7 a + 4\right)\cdot 13 + \left(9 a + 1\right)\cdot 13^{2} + \left(9 a + 8\right)\cdot 13^{3} + \left(7 a + 3\right)\cdot 13^{4} + \left(11 a + 10\right)\cdot 13^{5} + \left(6 a + 2\right)\cdot 13^{6} +O(13^{7})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 7 a + 11 + \left(10 a + 8\right)\cdot 13 + \left(3 a + 8\right)\cdot 13^{2} + 9\cdot 13^{3} + \left(3 a + 7\right)\cdot 13^{4} + \left(2 a + 7\right)\cdot 13^{5} + \left(12 a + 12\right)\cdot 13^{6} +O(13^{7})\)  Toggle raw display
$r_{ 3 }$ $=$ \( a + 5 + 5 a\cdot 13 + \left(3 a + 3\right)\cdot 13^{2} + \left(3 a + 8\right)\cdot 13^{3} + \left(5 a + 1\right)\cdot 13^{4} + \left(a + 1\right)\cdot 13^{5} + \left(6 a + 11\right)\cdot 13^{6} +O(13^{7})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 6 a + 5 + \left(2 a + 12\right)\cdot 13 + \left(9 a + 1\right)\cdot 13^{2} + \left(12 a + 6\right)\cdot 13^{3} + \left(9 a + 10\right)\cdot 13^{4} + \left(10 a + 6\right)\cdot 13^{5} + 9\cdot 13^{6} +O(13^{7})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 8 a + 3 + \left(2 a + 9\right)\cdot 13 + \left(7 a + 9\right)\cdot 13^{2} + \left(3 a + 11\right)\cdot 13^{3} + \left(8 a + 11\right)\cdot 13^{4} + \left(3 a + 8\right)\cdot 13^{5} + 5 a\cdot 13^{6} +O(13^{7})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 5 a + 11 + \left(10 a + 3\right)\cdot 13 + \left(5 a + 1\right)\cdot 13^{2} + \left(9 a + 8\right)\cdot 13^{3} + \left(4 a + 3\right)\cdot 13^{4} + \left(9 a + 4\right)\cdot 13^{5} + \left(7 a + 2\right)\cdot 13^{6} +O(13^{7})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.