Properties

Label 2.216.12t18.a.b
Dimension $2$
Group $C_6\times S_3$
Conductor $216$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $C_6\times S_3$
Conductor: \(216\)\(\medspace = 2^{3} \cdot 3^{3}\)
Artin stem field: 12.0.139314069504.1
Galois orbit size: $2$
Smallest permutation container: $C_6\times S_3$
Parity: odd
Determinant: 1.72.6t1.a.a
Projective image: S_3
Projective stem field: 3.1.648.1

Defining polynomial

$f(x)$$=$\(x^{12} - 2 x^{11} + x^{10} - 2 x^{9} + 3 x^{8} + 2 x^{7} - 6 x^{5} + x^{4} + 4 x^{3} - 2 x^{2} + 1\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \(x^{6} + 10 x^{3} + 11 x^{2} + 11 x + 2\)  Toggle raw display

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

Generators of the action on the roots $r_1, \ldots, r_{ 12 }$

Cycle notation
$(1,10)(2,11)(3,12)(4,7)(5,8)(6,9)$
$(1,9,8)(2,7,3)(4,12,11)(5,10,6)$
$(2,3,7)(4,11,12)$
$(1,11)(2,10)(3,5)(4,9)(6,7)(8,12)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 12 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,10)(2,11)(3,12)(4,7)(5,8)(6,9)$$-2$
$3$$2$$(1,11)(2,10)(3,5)(4,9)(6,7)(8,12)$$0$
$3$$2$$(1,2)(3,8)(4,6)(5,12)(7,9)(10,11)$$0$
$1$$3$$(1,9,8)(2,7,3)(4,12,11)(5,10,6)$$-2 \zeta_{3} - 2$
$1$$3$$(1,8,9)(2,3,7)(4,11,12)(5,6,10)$$2 \zeta_{3}$
$2$$3$$(2,3,7)(4,11,12)$$\zeta_{3} + 1$
$2$$3$$(2,7,3)(4,12,11)$$-\zeta_{3}$
$2$$3$$(1,8,9)(2,7,3)(4,12,11)(5,6,10)$$-1$
$1$$6$$(1,6,8,10,9,5)(2,4,3,11,7,12)$$2 \zeta_{3} + 2$
$1$$6$$(1,5,9,10,8,6)(2,12,7,11,3,4)$$-2 \zeta_{3}$
$2$$6$$(1,10)(2,12,7,11,3,4)(5,8)(6,9)$$-\zeta_{3} - 1$
$2$$6$$(1,10)(2,4,3,11,7,12)(5,8)(6,9)$$\zeta_{3}$
$2$$6$$(1,5,9,10,8,6)(2,4,3,11,7,12)$$1$
$3$$6$$(1,4,8,11,9,12)(2,6,3,10,7,5)$$0$
$3$$6$$(1,12,9,11,8,4)(2,5,7,10,3,6)$$0$
$3$$6$$(1,7,8,2,9,3)(4,5,11,6,12,10)$$0$
$3$$6$$(1,3,9,2,8,7)(4,10,12,6,11,5)$$0$

The blue line marks the conjugacy class containing complex conjugation.