Properties

Label 2.2520.12t18.c.a
Dimension $2$
Group $C_6\times S_3$
Conductor $2520$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $C_6\times S_3$
Conductor: \(2520\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Artin stem field: 12.0.7169347584000000.2
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)$$=$\(x^{12} - 2 x^{11} + 3 x^{10} + 2 x^{9} - 15 x^{8} + 26 x^{7} - 26 x^{6} - 6 x^{5} + 93 x^{4} - 196 x^{3} + 268 x^{2} - 220 x + 121\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 7.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \(x^{6} + 2 x^{4} + 10 x^{2} + 3 x + 3\)  Toggle raw display

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.