Properties

Label 2.1352.12t18.a.b
Dimension $2$
Group $C_6\times S_3$
Conductor $1352$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $C_6\times S_3$
Conductor: \(1352\)\(\medspace = 2^{3} \cdot 13^{2}\)
Artin stem field: Galois closure of 12.0.36138776487264256.6
Galois orbit size: $2$
Smallest permutation container: $C_6\times S_3$
Parity: odd
Determinant: 1.104.6t1.b.a
Projective image: $S_3$
Projective stem field: Galois closure of 3.1.104.1

Defining polynomial

$f(x)$$=$ \( x^{12} - 2 x^{11} + 13 x^{10} - 10 x^{9} + 61 x^{8} - 30 x^{7} + 202 x^{6} - 74 x^{5} + 347 x^{4} + 36 x^{3} + 130 x^{2} + 132 x + 27 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 8.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: \( x^{6} + 17x^{3} + 17x^{2} + 6x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 4 a^{5} + 16 a^{4} + 8 a^{3} + 8 a^{2} + 7 a + 14 + \left(7 a^{5} + 5 a^{4} + 5 a^{3} + 4 a^{2} + a + 12\right)\cdot 19 + \left(6 a^{5} + 12 a^{4} + 2 a^{3} + 18 a^{2} + 14\right)\cdot 19^{2} + \left(8 a^{5} + 18 a^{4} + 3 a^{3} + 10 a^{2} + 13 a + 4\right)\cdot 19^{3} + \left(7 a^{4} + 12 a^{2} + 11 a + 12\right)\cdot 19^{4} + \left(2 a^{5} + 4 a^{4} + 10 a^{3} + 2 a^{2} + 18 a\right)\cdot 19^{5} + \left(12 a^{5} + 10 a^{4} + 2 a^{3} + 10 a^{2} + 5\right)\cdot 19^{6} + \left(17 a^{5} + 3 a^{4} + 3 a^{3} + 9 a^{2} + 16 a + 1\right)\cdot 19^{7} +O(19^{8})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 9 a^{5} + 8 a^{4} + 11 a^{3} + a^{2} + 8 a + 13 + \left(6 a^{4} + 7 a^{3} + 10 a^{2} + 10 a + 18\right)\cdot 19 + \left(11 a^{5} + 10 a^{4} + 4 a^{3} + 11 a^{2} + 15 a + 8\right)\cdot 19^{2} + \left(4 a^{5} + 18 a^{4} + 14 a^{3} + 4 a^{2} + 17 a + 14\right)\cdot 19^{3} + \left(2 a^{5} + 16 a^{4} + 16 a^{3} + 6 a^{2} + 14 a + 9\right)\cdot 19^{4} + \left(15 a^{5} + 10 a^{4} + a^{3} + 8 a^{2} + 14 a + 14\right)\cdot 19^{5} + \left(9 a^{5} + 3 a^{4} + 17 a^{3} + 17 a^{2} + 8 a + 9\right)\cdot 19^{6} + \left(2 a^{5} + 4 a^{4} + 11 a^{3} + 6 a^{2} + 18 a + 4\right)\cdot 19^{7} +O(19^{8})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 16 a^{5} + 13 a^{4} + 7 a^{3} + 6 a^{2} + 17 a + 1 + \left(14 a^{5} + 7 a^{4} + 16 a^{3} + 11 a^{2} + 16 a + 18\right)\cdot 19 + \left(13 a^{5} + 4 a^{4} + 8 a^{3} + 13 a^{2} + 4 a + 2\right)\cdot 19^{2} + \left(13 a^{5} + 7 a^{4} + a^{3} + 17 a^{2} + a + 7\right)\cdot 19^{3} + \left(14 a^{5} + 14 a^{4} + 15 a^{3} + 9 a + 6\right)\cdot 19^{4} + \left(4 a^{5} + 12 a^{4} + 4 a^{3} + 9 a^{2} + 6 a + 2\right)\cdot 19^{5} + \left(11 a^{5} + 2 a^{4} + 5 a^{3} + 12 a^{2} + 6 a + 1\right)\cdot 19^{6} + \left(a^{5} + 3 a^{4} + 14 a^{3} + 17 a^{2} + 8 a + 5\right)\cdot 19^{7} +O(19^{8})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 8 a^{5} + 9 a^{4} + 15 a^{3} + 9 a^{2} + 6 a + 11 + \left(16 a^{5} + 3 a^{4} + 2 a^{3} + a^{2} + 5 a + 9\right)\cdot 19 + \left(4 a^{5} + a^{4} + 16 a^{3} + 16 a^{2} + 5 a + 11\right)\cdot 19^{2} + \left(16 a^{5} + 16 a^{4} + 10 a^{2} + 11 a + 11\right)\cdot 19^{3} + \left(8 a^{5} + 12 a^{4} + 2 a^{3} + 18 a^{2} + 2 a + 14\right)\cdot 19^{4} + \left(6 a^{5} + 15 a^{4} + 17 a^{3} + 12 a^{2} + 14 a + 1\right)\cdot 19^{5} + \left(12 a^{5} + 6 a^{4} + 2 a^{3} + 8 a^{2} + 8 a + 15\right)\cdot 19^{6} + \left(a^{5} + 5 a^{4} + 13 a^{3} + 9 a^{2} + 15 a + 1\right)\cdot 19^{7} +O(19^{8})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( a^{4} + 11 a^{3} + 12 a^{2} + 15 a + 11 + \left(5 a^{5} + 18 a^{4} + 5 a^{3} + 7 a^{2} + 13 a + 11\right)\cdot 19 + \left(8 a^{5} + 12 a^{4} + 7 a^{3} + 17 a^{2} + 4 a + 10\right)\cdot 19^{2} + \left(17 a^{5} + 10 a^{4} + 12 a^{3} + 2 a^{2} + 3 a + 16\right)\cdot 19^{3} + \left(18 a^{5} + 16 a^{4} + 3 a^{3} + 14 a^{2} + 2 a + 9\right)\cdot 19^{4} + \left(14 a^{5} + 4 a^{4} + 16 a^{3} + 9 a^{2} + 6 a + 15\right)\cdot 19^{5} + \left(7 a^{5} + 7 a^{4} + 2 a^{3} + 2 a^{2} + 13 a + 15\right)\cdot 19^{6} + \left(7 a^{5} + 13 a^{4} + 13 a^{3} + 14 a^{2} + 9 a + 13\right)\cdot 19^{7} +O(19^{8})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 17 a^{5} + 16 a^{4} + 10 a^{3} + 10 a^{2} + 7 a + 18 + \left(14 a^{5} + 18 a^{4} + 7 a^{2} + 8 a + 6\right)\cdot 19 + \left(18 a^{5} + 18 a^{4} + 18 a^{3} + 8 a^{2} + 9 a + 8\right)\cdot 19^{2} + \left(9 a^{5} + 2 a^{4} + 18 a^{3} + 15 a^{2} + 4 a\right)\cdot 19^{3} + \left(8 a^{5} + a^{4} + 15 a^{3} + a^{2} + 10 a + 17\right)\cdot 19^{4} + \left(13 a^{5} + 3 a^{4} + 4 a^{3} + 12 a^{2} + a + 5\right)\cdot 19^{5} + \left(17 a^{5} + 10 a^{4} + 12 a^{3} + 9 a^{2} + 17 a + 16\right)\cdot 19^{6} + \left(9 a^{5} + 13 a^{4} + 7 a^{3} + 3 a^{2} + 7 a + 6\right)\cdot 19^{7} +O(19^{8})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( a^{5} + 4 a^{4} + 2 a^{3} + 7 a^{2} + a + 2 + \left(a^{5} + 11 a^{3} + 4 a^{2} + 9 a + 17\right)\cdot 19 + \left(14 a^{5} + 3 a^{4} + 13 a^{3} + 9 a^{2} + 8 a + 8\right)\cdot 19^{2} + \left(4 a^{5} + 14 a^{4} + 16 a^{3} + 17 a + 13\right)\cdot 19^{3} + \left(9 a^{5} + 18 a^{4} + a^{3} + 9 a^{2} + 6 a + 18\right)\cdot 19^{4} + \left(7 a^{5} + 14 a^{4} + 16 a^{3} + 11 a^{2} + 4 a + 18\right)\cdot 19^{5} + \left(14 a^{5} + 13 a^{3} + 14 a^{2} + 11 a + 11\right)\cdot 19^{6} + \left(11 a^{5} + 16 a^{4} + 16 a^{3} + 13 a^{2} + 7 a + 10\right)\cdot 19^{7} +O(19^{8})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 7 a^{5} + 10 a^{4} + 11 a^{3} + 18 a^{2} + 9 a + 13 + \left(10 a^{5} + 16 a^{3} + 4 a^{2} + 6 a + 17\right)\cdot 19 + \left(7 a^{5} + 10 a^{4} + a^{2} + 5 a + 11\right)\cdot 19^{2} + \left(10 a^{5} + 18 a^{4} + 7 a^{3} + 8 a^{2} + 14 a + 7\right)\cdot 19^{3} + \left(15 a^{5} + 8 a^{4} + 7 a^{3} + 18 a^{2} + 7 a + 10\right)\cdot 19^{4} + \left(15 a^{5} + 16 a^{4} + 18 a^{3} + 6 a^{2} + 15 a + 12\right)\cdot 19^{5} + \left(10 a^{5} + 3 a^{4} + 3 a^{3} + 13 a^{2} + 14 a + 1\right)\cdot 19^{6} + \left(14 a^{5} + a^{4} + 5 a^{3} + 18 a^{2} + 8 a + 12\right)\cdot 19^{7} +O(19^{8})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 10 a^{5} + 9 a^{4} + 9 a^{3} + 17 a^{2} + 2 a + 6 + \left(13 a^{5} + 8 a^{4} + 6 a^{3} + 2 a^{2} + 4 a + 12\right)\cdot 19 + \left(16 a^{5} + 7 a^{4} + 15 a^{3} + 17 a^{2} + 8 a + 2\right)\cdot 19^{2} + \left(7 a^{5} + 18 a^{4} + 4 a^{3} + 14 a^{2} + 13 a + 6\right)\cdot 19^{3} + \left(15 a^{5} + 13 a^{4} + 16 a^{3} + 11 a^{2} + 7 a + 18\right)\cdot 19^{4} + \left(12 a^{5} + 18 a^{4} + 18 a^{3} + 8 a^{2} + 13 a + 4\right)\cdot 19^{5} + \left(4 a^{5} + 13 a^{4} + 14 a^{3} + a^{2} + 8\right)\cdot 19^{6} + \left(7 a^{5} + 18 a^{4} + 5 a^{3} + 6 a^{2} + 12 a + 14\right)\cdot 19^{7} +O(19^{8})\) Copy content Toggle raw display
$r_{ 10 }$ $=$ \( 11 a^{5} + a^{3} + 7 a^{2} + 7 a + 14 + \left(13 a^{5} + 13 a^{4} + 13 a^{2} + 17 a + 16\right)\cdot 19 + \left(a^{5} + 13 a^{4} + 17 a^{3} + 11 a^{2} + 12 a + 14\right)\cdot 19^{2} + \left(2 a^{5} + 18 a^{4} + 9 a^{3} + 7 a^{2} + 16 a + 9\right)\cdot 19^{3} + \left(2 a^{5} + a^{4} + 17 a^{3} + 7 a^{2} + 5 a + 11\right)\cdot 19^{4} + \left(5 a^{5} + 5 a^{4} + 14 a^{3} + 13 a^{2} + 16 a + 8\right)\cdot 19^{5} + \left(6 a^{5} + 11 a^{4} + 2 a^{3} + 6 a^{2} + 8 a + 2\right)\cdot 19^{6} + \left(13 a^{5} + 18 a^{4} + 18 a^{3} + 11 a^{2} + 13 a + 15\right)\cdot 19^{7} +O(19^{8})\) Copy content Toggle raw display
$r_{ 11 }$ $=$ \( 14 a^{5} + 8 a^{4} + a^{3} + 14 a^{2} + 2 a + 12 + \left(13 a^{5} + 16 a^{4} + 13 a^{3} + 6 a^{2} + 10 a + 11\right)\cdot 19 + \left(2 a^{5} + 13 a^{4} + 3 a^{2} + 6 a + 3\right)\cdot 19^{2} + \left(8 a^{5} + 16 a^{4} + 14 a^{3} + 5 a^{2} + 14 a + 5\right)\cdot 19^{3} + \left(17 a^{5} + 17 a^{4} + 12 a^{3} + 14 a^{2} + 8 a + 12\right)\cdot 19^{4} + \left(a^{5} + 11 a^{4} + a^{3} + 6 a^{2} + 16 a + 17\right)\cdot 19^{5} + \left(4 a^{5} + a^{4} + 13 a^{3} + 14 a^{2} + 13 a + 7\right)\cdot 19^{6} + \left(5 a^{5} + 11 a^{3} + 17 a^{2} + 13 a + 4\right)\cdot 19^{7} +O(19^{8})\) Copy content Toggle raw display
$r_{ 12 }$ $=$ \( 17 a^{5} + a^{4} + 9 a^{3} + 5 a^{2} + 14 a + 1 + \left(2 a^{5} + 15 a^{4} + 9 a^{3} + a^{2} + 10 a + 18\right)\cdot 19 + \left(8 a^{5} + 5 a^{4} + 9 a^{3} + 5 a^{2} + 13 a + 14\right)\cdot 19^{2} + \left(10 a^{5} + 10 a^{4} + 10 a^{3} + 15 a^{2} + 5 a + 16\right)\cdot 19^{3} + \left(a^{4} + 4 a^{3} + 17 a^{2} + 7 a + 10\right)\cdot 19^{4} + \left(14 a^{5} + 14 a^{4} + 8 a^{3} + 11 a^{2} + 5 a + 10\right)\cdot 19^{5} + \left(2 a^{5} + 3 a^{4} + 3 a^{3} + 2 a^{2} + 9 a + 18\right)\cdot 19^{6} + \left(2 a^{5} + 16 a^{4} + 12 a^{3} + 4 a^{2} + a + 4\right)\cdot 19^{7} +O(19^{8})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.