Properties

Label 2.672.12t18.c.b
Dimension $2$
Group $C_6\times S_3$
Conductor $672$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $C_6\times S_3$
Conductor: \(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \)
Artin stem field: Galois closure of 12.0.29365647704064.2
Galois orbit size: $2$
Smallest permutation container: $C_6\times S_3$
Parity: odd
Determinant: 1.168.6t1.c.a
Projective image: $S_3$
Projective stem field: Galois closure of 3.1.1176.1

Defining polynomial

$f(x)$$=$ \( x^{12} - 4 x^{11} + 10 x^{10} - 16 x^{9} + 20 x^{8} - 20 x^{7} + 18 x^{6} - 20 x^{5} + 7 x^{4} + 4 x^{3} + 4 x^{2} - 4 x + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 7.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \( x^{6} + 2x^{4} + 10x^{2} + 3x + 3 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 16 a^{5} + 12 a^{4} + 4 a^{3} + 13 a^{2} + 13 a + 6 + \left(15 a^{5} + 2 a^{4} + 13 a^{3} + 9 a^{2} + 8 a + 13\right)\cdot 17 + \left(5 a^{5} + 4 a^{4} + 11 a^{3} + 12 a^{2} + 5 a + 11\right)\cdot 17^{2} + \left(a^{5} + 12 a^{4} + 11 a^{3} + 7 a^{2} + 5 a + 14\right)\cdot 17^{3} + \left(13 a^{5} + 5 a^{4} + 16 a^{3} + 5 a^{2} + 9 a + 11\right)\cdot 17^{4} + \left(3 a^{5} + a^{4} + 6 a^{3} + 5 a + 5\right)\cdot 17^{5} + \left(7 a^{5} + 5 a^{4} + 5 a^{3} + 11 a^{2} + 4 a + 12\right)\cdot 17^{6} +O(17^{7})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 16 a^{5} + 12 a^{4} + 4 a^{3} + 13 a^{2} + 13 a + 16 + \left(15 a^{5} + 2 a^{4} + 13 a^{3} + 9 a^{2} + 8 a + 11\right)\cdot 17 + \left(5 a^{5} + 4 a^{4} + 11 a^{3} + 12 a^{2} + 5 a + 16\right)\cdot 17^{2} + \left(a^{5} + 12 a^{4} + 11 a^{3} + 7 a^{2} + 5 a + 13\right)\cdot 17^{3} + \left(13 a^{5} + 5 a^{4} + 16 a^{3} + 5 a^{2} + 9 a + 2\right)\cdot 17^{4} + \left(3 a^{5} + a^{4} + 6 a^{3} + 5 a + 5\right)\cdot 17^{5} + \left(7 a^{5} + 5 a^{4} + 5 a^{3} + 11 a^{2} + 4 a + 13\right)\cdot 17^{6} +O(17^{7})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( a^{5} + 4 a^{3} + 8 a^{2} + 4 a + 16 + \left(3 a^{5} + 2 a^{4} + 12 a^{3} + 15 a^{2} + 9 a + 10\right)\cdot 17 + \left(a^{5} + 3 a^{4} + 4 a^{3} + 14 a^{2} + 11 a + 3\right)\cdot 17^{2} + \left(10 a^{5} + a^{4} + 9 a^{3} + 14 a^{2} + 12 a + 4\right)\cdot 17^{3} + \left(12 a^{5} + 9 a^{4} + 6 a^{3} + a^{2} + 10 a + 14\right)\cdot 17^{4} + \left(7 a^{5} + a^{4} + 15 a^{3} + 2 a^{2} + 10 a + 8\right)\cdot 17^{5} + \left(6 a^{5} + 14 a^{4} + 2 a^{3} + 5 a^{2} + 7 a + 4\right)\cdot 17^{6} +O(17^{7})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( a^{5} + 4 a^{3} + 8 a^{2} + 4 a + 6 + \left(3 a^{5} + 2 a^{4} + 12 a^{3} + 15 a^{2} + 9 a + 12\right)\cdot 17 + \left(a^{5} + 3 a^{4} + 4 a^{3} + 14 a^{2} + 11 a + 15\right)\cdot 17^{2} + \left(10 a^{5} + a^{4} + 9 a^{3} + 14 a^{2} + 12 a + 4\right)\cdot 17^{3} + \left(12 a^{5} + 9 a^{4} + 6 a^{3} + a^{2} + 10 a + 6\right)\cdot 17^{4} + \left(7 a^{5} + a^{4} + 15 a^{3} + 2 a^{2} + 10 a + 9\right)\cdot 17^{5} + \left(6 a^{5} + 14 a^{4} + 2 a^{3} + 5 a^{2} + 7 a + 3\right)\cdot 17^{6} +O(17^{7})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 9 a^{5} + 4 a^{3} + 15 a^{2} + 9 a + 1 + \left(10 a^{5} + 16 a^{4} + 10 a^{3} + 8 a^{2} + 5 a + 12\right)\cdot 17 + \left(12 a^{5} + 14 a^{4} + 2 a^{3} + 12 a^{2} + 10 a + 8\right)\cdot 17^{2} + \left(6 a^{5} + 13 a^{4} + 3 a^{3} + 5 a^{2} + 5 a + 6\right)\cdot 17^{3} + \left(16 a^{5} + 16 a^{4} + 2 a^{3} + 7 a^{2} + 3 a + 6\right)\cdot 17^{4} + \left(8 a^{5} + 12 a^{4} + 7 a^{2} + 11\right)\cdot 17^{5} + \left(9 a^{5} + 7 a^{4} + 16 a^{3} + 11 a^{2} + 16\right)\cdot 17^{6} +O(17^{7})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 9 a^{5} + 4 a^{3} + 15 a^{2} + 9 a + 8 + \left(10 a^{5} + 16 a^{4} + 10 a^{3} + 8 a^{2} + 5 a + 13\right)\cdot 17 + \left(12 a^{5} + 14 a^{4} + 2 a^{3} + 12 a^{2} + 10 a + 3\right)\cdot 17^{2} + \left(6 a^{5} + 13 a^{4} + 3 a^{3} + 5 a^{2} + 5 a + 7\right)\cdot 17^{3} + \left(16 a^{5} + 16 a^{4} + 2 a^{3} + 7 a^{2} + 3 a + 15\right)\cdot 17^{4} + \left(8 a^{5} + 12 a^{4} + 7 a^{2} + 11\right)\cdot 17^{5} + \left(9 a^{5} + 7 a^{4} + 16 a^{3} + 11 a^{2} + 15\right)\cdot 17^{6} +O(17^{7})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 16 a^{5} + 12 a^{4} + 8 a^{3} + 9 a^{2} + 2 a + 9 + \left(6 a^{5} + 4 a^{4} + a^{3} + 11 a^{2} + 13 a + 5\right)\cdot 17 + \left(5 a^{4} + 15 a^{2} + 12 a\right)\cdot 17^{2} + \left(12 a^{5} + 6 a^{4} + 3 a^{3} + 3 a^{2} + 9 a + 10\right)\cdot 17^{3} + \left(3 a^{5} + 9 a^{4} + 11 a^{3} + 10 a^{2} + 2 a + 2\right)\cdot 17^{4} + \left(4 a^{5} + 3 a^{4} + 16 a^{3} + 9 a^{2} + a + 2\right)\cdot 17^{5} + \left(7 a^{5} + 2 a^{4} + 11 a^{3} + 3 a^{2} + 10 a + 9\right)\cdot 17^{6} +O(17^{7})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 16 a^{5} + 12 a^{4} + 8 a^{3} + 9 a^{2} + 2 a + 2 + \left(6 a^{5} + 4 a^{4} + a^{3} + 11 a^{2} + 13 a + 4\right)\cdot 17 + \left(5 a^{4} + 15 a^{2} + 12 a + 5\right)\cdot 17^{2} + \left(12 a^{5} + 6 a^{4} + 3 a^{3} + 3 a^{2} + 9 a + 9\right)\cdot 17^{3} + \left(3 a^{5} + 9 a^{4} + 11 a^{3} + 10 a^{2} + 2 a + 10\right)\cdot 17^{4} + \left(4 a^{5} + 3 a^{4} + 16 a^{3} + 9 a^{2} + a + 1\right)\cdot 17^{5} + \left(7 a^{5} + 2 a^{4} + 11 a^{3} + 3 a^{2} + 10 a + 10\right)\cdot 17^{6} +O(17^{7})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 4 a^{5} + 5 a^{4} + 14 a^{3} + 16 a^{2} + 6 a + 3 + \left(9 a^{5} + 14 a^{4} + 11 a^{3} + 14 a^{2} + 5 a + 5\right)\cdot 17 + \left(5 a^{5} + 2 a^{4} + 6 a^{3} + 8 a^{2} + 9 a + 1\right)\cdot 17^{2} + \left(16 a^{5} + 2 a^{3} + 8 a^{2} + 10 a + 8\right)\cdot 17^{3} + \left(2 a^{5} + 14 a^{4} + 10 a^{3} + 4 a^{2} + a + 7\right)\cdot 17^{4} + \left(7 a^{5} + 14 a^{4} + 8 a^{3} + 9 a^{2} + 14 a + 1\right)\cdot 17^{5} + \left(8 a^{5} + 11 a^{4} + 12 a^{3} + 6 a^{2} + 14 a + 6\right)\cdot 17^{6} +O(17^{7})\) Copy content Toggle raw display
$r_{ 10 }$ $=$ \( 4 a^{5} + 5 a^{4} + 14 a^{3} + 16 a^{2} + 6 a + 13 + \left(9 a^{5} + 14 a^{4} + 11 a^{3} + 14 a^{2} + 5 a + 3\right)\cdot 17 + \left(5 a^{5} + 2 a^{4} + 6 a^{3} + 8 a^{2} + 9 a + 6\right)\cdot 17^{2} + \left(16 a^{5} + 2 a^{3} + 8 a^{2} + 10 a + 7\right)\cdot 17^{3} + \left(2 a^{5} + 14 a^{4} + 10 a^{3} + 4 a^{2} + a + 15\right)\cdot 17^{4} + \left(7 a^{5} + 14 a^{4} + 8 a^{3} + 9 a^{2} + 14 a\right)\cdot 17^{5} + \left(8 a^{5} + 11 a^{4} + 12 a^{3} + 6 a^{2} + 14 a + 7\right)\cdot 17^{6} +O(17^{7})\) Copy content Toggle raw display
$r_{ 11 }$ $=$ \( 5 a^{5} + 5 a^{4} + 7 a^{2} + 1 + \left(5 a^{5} + 11 a^{4} + 2 a^{3} + 7 a^{2} + 9 a + 4\right)\cdot 17 + \left(8 a^{5} + 3 a^{4} + 8 a^{3} + 3 a^{2} + a + 8\right)\cdot 17^{2} + \left(4 a^{5} + 4 a^{3} + 10 a^{2} + 7 a + 7\right)\cdot 17^{3} + \left(2 a^{5} + 13 a^{4} + 4 a^{3} + 4 a^{2} + 6 a + 8\right)\cdot 17^{4} + \left(2 a^{5} + 16 a^{4} + 3 a^{3} + 5 a^{2} + 2 a + 4\right)\cdot 17^{5} + \left(12 a^{5} + 9 a^{4} + 2 a^{3} + 13 a^{2} + 14 a + 2\right)\cdot 17^{6} +O(17^{7})\) Copy content Toggle raw display
$r_{ 12 }$ $=$ \( 5 a^{5} + 5 a^{4} + 7 a^{2} + 8 + \left(5 a^{5} + 11 a^{4} + 2 a^{3} + 7 a^{2} + 9 a + 5\right)\cdot 17 + \left(8 a^{5} + 3 a^{4} + 8 a^{3} + 3 a^{2} + a + 3\right)\cdot 17^{2} + \left(4 a^{5} + 4 a^{3} + 10 a^{2} + 7 a + 8\right)\cdot 17^{3} + \left(2 a^{5} + 13 a^{4} + 4 a^{3} + 4 a^{2} + 6 a\right)\cdot 17^{4} + \left(2 a^{5} + 16 a^{4} + 3 a^{3} + 5 a^{2} + 2 a + 5\right)\cdot 17^{5} + \left(12 a^{5} + 9 a^{4} + 2 a^{3} + 13 a^{2} + 14 a + 1\right)\cdot 17^{6} +O(17^{7})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.