Properties

Label 2.455.12t18.a.a
Dimension $2$
Group $C_6\times S_3$
Conductor $455$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $C_6\times S_3$
Conductor: \(455\)\(\medspace = 5 \cdot 7 \cdot 13 \)
Artin stem field: 12.0.52502704515625.1
Galois orbit size: $2$
Smallest permutation container: $C_6\times S_3$
Parity: odd
Determinant: 1.455.6t1.b.b
Projective image: $S_3$
Projective stem field: 3.1.5915.1

Defining polynomial

$f(x)$$=$\(x^{12} - 3 x^{11} + 7 x^{10} - 11 x^{9} + 16 x^{8} - 10 x^{7} + 23 x^{6} + 10 x^{5} + 16 x^{4} + 11 x^{3} + 7 x^{2} + 3 x + 1\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 7.

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.