Properties

Label 1.45.12t1.b.c
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: \(\Q(\zeta_{45})^+\)
Galois orbit size: $4$
Smallest permutation container: $C_{12}$
Parity: even
Dirichlet character: \(\chi_{45}(23,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{12} - 12 x^{10} - x^{9} + 54 x^{8} + 9 x^{7} - 112 x^{6} - 27 x^{5} + 105 x^{4} + 31 x^{3} - 36 x^{2} - 12 x + 1\)  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:
$r_{ 1 }$ $=$ \( 14 a^{3} + 3 a^{2} + 2 a + 16 + \left(9 a^{3} + 4 a^{2} + 16 a + 14\right)\cdot 17 + \left(9 a^{3} + 3 a^{2} + 10 a + 3\right)\cdot 17^{2} + \left(11 a^{3} + 16 a^{2} + 5 a + 5\right)\cdot 17^{3} + \left(6 a^{3} + 12 a^{2} + a + 5\right)\cdot 17^{4} +O(17^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 8 a^{3} + 11 a^{2} + 7 a + 6 + \left(6 a^{3} + 16 a^{2} + 8 a + 11\right)\cdot 17 + \left(a^{2} + 5 a + 4\right)\cdot 17^{2} + \left(11 a^{3} + 14 a^{2} + 9 a + 7\right)\cdot 17^{3} + \left(14 a^{3} + 3 a^{2} + 4 a + 12\right)\cdot 17^{4} +O(17^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 7 a^{3} + 6 a^{2} + 13 a + 8 + \left(13 a^{2} + 11 a + 10\right)\cdot 17 + \left(14 a^{3} + 4 a^{2} + 8 a + 8\right)\cdot 17^{2} + \left(12 a^{3} + 14 a^{2} + 5 a + 16\right)\cdot 17^{3} + \left(6 a^{3} + 11 a^{2} + a + 1\right)\cdot 17^{4} +O(17^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 4 a^{3} + 16 a^{2} + 13 a + 12 + \left(5 a^{3} + 3 a^{2} + 16 a + 13\right)\cdot 17 + \left(9 a^{3} + a^{2} + 3 a + 11\right)\cdot 17^{2} + \left(9 a^{3} + 4 a^{2} + 14 a + 15\right)\cdot 17^{3} + \left(15 a^{3} + 10 a^{2} + 5 a + 2\right)\cdot 17^{4} +O(17^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 14 a^{3} + 14 a^{2} + 16 a + 6 + \left(4 a^{3} + 13 a^{2} + 11 a + 15\right)\cdot 17 + \left(a^{3} + 2 a^{2} + 10 a + 1\right)\cdot 17^{2} + \left(4 a^{3} + 12 a^{2} + 10 a + 13\right)\cdot 17^{3} + \left(11 a^{3} + a^{2} + 10\right)\cdot 17^{4} +O(17^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 6 a^{3} + 16 a + 12 + \left(2 a^{3} + 16 a^{2} + 5 a + 3\right)\cdot 17 + \left(6 a^{3} + 10 a^{2} + 12 a + 11\right)\cdot 17^{2} + \left(a^{3} + 5 a^{2} + 15\right)\cdot 17^{3} + \left(16 a^{3} + 2 a^{2} + 15 a\right)\cdot 17^{4} +O(17^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 3 a^{3} + a^{2} + 13 a + 10 + \left(13 a^{3} + 6 a^{2} + 8 a + 7\right)\cdot 17 + \left(11 a^{3} + 2 a^{2} + 6\right)\cdot 17^{2} + \left(7 a^{3} + 3 a^{2} + 13 a + 3\right)\cdot 17^{3} + \left(3 a^{3} + 3 a^{2} + 15 a + 11\right)\cdot 17^{4} +O(17^{5})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 10 a^{3} + 6 a^{2} + 6 a + 16 + \left(4 a^{3} + 8 a^{2} + 11 a + 2\right)\cdot 17 + \left(4 a^{3} + 10 a^{2} + 9 a\right)\cdot 17^{2} + \left(7 a^{3} + 8 a^{2} + 16\right)\cdot 17^{3} + \left(10 a^{3} + 14 a^{2} + 6\right)\cdot 17^{4} +O(17^{5})\)  Toggle raw display
$r_{ 9 }$ $=$ \( 5 a^{3} + 7 a^{2} + 14 a + 16 + \left(5 a^{3} + 13 a^{2} + 8 a + 8\right)\cdot 17 + \left(7 a^{3} + 13 a^{2} + 7 a\right)\cdot 17^{2} + \left(13 a^{3} + 15 a^{2} + 10 a + 11\right)\cdot 17^{3} + \left(3 a^{3} + 2 a^{2} + 6 a + 1\right)\cdot 17^{4} +O(17^{5})\)  Toggle raw display
$r_{ 10 }$ $=$ \( 9 a^{3} + 9 a^{2} + 6 a + 8 + \left(a^{3} + 12 a^{2} + 6 a + 7\right)\cdot 17 + \left(a^{3} + 7 a^{2} + 10 a + 15\right)\cdot 17^{2} + \left(16 a^{2} + 8 a + 4\right)\cdot 17^{3} + \left(5 a^{3} + 15 a^{2} + 8 a + 3\right)\cdot 17^{4} +O(17^{5})\)  Toggle raw display
$r_{ 11 }$ $=$ \( 5 a^{3} + 7 a^{2} + 15 a + 16 + \left(2 a^{3} + 15 a^{2} + a + 1\right)\cdot 17 + \left(4 a^{3} + 6 a^{2} + 6 a + 12\right)\cdot 17^{2} + \left(9 a^{3} + 14 a^{2} + 12 a + 8\right)\cdot 17^{3} + \left(8 a^{3} + 14 a^{2} + 9 a + 2\right)\cdot 17^{4} +O(17^{5})\)  Toggle raw display
$r_{ 12 }$ $=$ \( 5 a^{2} + 15 a + 10 + \left(12 a^{3} + 12 a^{2} + 10 a + 3\right)\cdot 17 + \left(15 a^{3} + a^{2} + 15 a + 8\right)\cdot 17^{2} + \left(13 a^{3} + 11 a^{2} + 10 a + 1\right)\cdot 17^{3} + \left(16 a^{3} + 7 a^{2} + 15 a + 8\right)\cdot 17^{4} +O(17^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.