Properties

Label 1.180.12t1.b.b
Dimension $1$
Group $C_{12}$
Conductor $180$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_{12}$
Conductor: \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \)
Artin field: 12.12.344373768000000000.1
Galois orbit size: $4$
Smallest permutation container: $C_{12}$
Parity: even
Dirichlet character: \(\chi_{180}(43,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{12} - 27 x^{10} - 4 x^{9} + 234 x^{8} + 36 x^{7} - 737 x^{6} + 72 x^{5} + 795 x^{4} - 336 x^{3} - 96 x^{2} + 42 x + 1\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \(x^{4} + 7 x^{2} + 10 x + 3\)  Toggle raw display

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.