Properties

Label 1.45.12t1.a.d
Dimension $1$
Group $C_{12}$
Conductor $45$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_{12}$
Conductor: \(45\)\(\medspace = 3^{2} \cdot 5 \)
Artin field: 12.0.84075626953125.1
Galois orbit size: $4$
Smallest permutation container: $C_{12}$
Parity: odd
Dirichlet character: \(\chi_{45}(13,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{12} + 3 x^{10} - x^{9} + 9 x^{8} + 9 x^{7} + 28 x^{6} + 18 x^{5} + 75 x^{4} + 26 x^{3} + 9 x^{2} + 3 x + 1\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.