Properties

Label 2.297.12t18.b.b
Dimension $2$
Group $C_6\times S_3$
Conductor $297$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $C_6\times S_3$
Conductor: \(297\)\(\medspace = 3^{3} \cdot 11 \)
Artin stem field: Galois closure of 12.0.941480149401.1
Galois orbit size: $2$
Smallest permutation container: $C_6\times S_3$
Parity: odd
Determinant: 1.99.6t1.a.a
Projective image: $S_3$
Projective stem field: Galois closure of 3.1.891.1

Defining polynomial

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.