Properties

Label 2.3920.12t18.b.a
Dimension $2$
Group $C_6\times S_3$
Conductor $3920$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $C_6\times S_3$
Conductor: \(3920\)\(\medspace = 2^{4} \cdot 5 \cdot 7^{2}\)
Artin stem field: 12.0.289254654976000000.2
Galois orbit size: $2$
Smallest permutation container: $C_6\times S_3$
Parity: odd
Determinant: 1.35.6t1.a.b
Projective image: $S_3$
Projective stem field: 3.1.140.1

Defining polynomial

$f(x)$$=$\(x^{12} - 2 x^{11} + 2 x^{10} - 14 x^{9} - 60 x^{8} + 36 x^{7} + 146 x^{6} + 268 x^{5} + 936 x^{4} + 1316 x^{3} + 1152 x^{2} + 1056 x + 484\)  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 $\mathbb{Q}_{ 19 }$: \(x^{6} + 17 x^{3} + 17 x^{2} + 6 x + 2\)  Toggle raw display

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.