Properties

Label 2.280.12t18.b.a
Dimension $2$
Group $C_6\times S_3$
Conductor $280$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $C_6\times S_3$
Conductor: \(280\)\(\medspace = 2^{3} \cdot 5 \cdot 7 \)
Artin stem field: 12.0.9834496000000.1
Galois orbit size: $2$
Smallest permutation container: $C_6\times S_3$
Parity: odd
Determinant: 1.280.6t1.d.b
Projective image: $S_3$
Projective stem field: 3.1.1960.1

Defining polynomial

$f(x)$$=$\(x^{12} - 2 x^{11} - x^{10} - 2 x^{9} + 9 x^{8} + 2 x^{7} - 4 x^{6} - 8 x^{5} + 2 x^{4} + 8 x^{3} + 7 x^{2} - 2 x + 1\)  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 $\mathbb{Q}_{ 13 }$: \(x^{6} + 10 x^{3} + 11 x^{2} + 11 x + 2\)  Toggle raw display

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.