Properties

Label 1.65.12t1.a
Dimension $1$
Group $C_{12}$
Conductor $65$
Indicator $0$

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Basic invariants

Dimension:$1$
Group:$C_{12}$
Conductor:\(65\)\(\medspace = 5 \cdot 13 \)
Artin number field: Galois closure of 12.0.1593224064453125.1
Galois orbit size: $4$
Smallest permutation container: $C_{12}$
Parity: odd
Projective image: $C_1$
Projective field: \(\Q\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: \(x^{4} + 8 x^{2} + 40 x + 5\)  Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 39 a^{3} + 5 a^{2} + 14 a + 29 + \left(5 a^{3} + 38 a^{2} + 45 a + 24\right)\cdot 47 + \left(41 a^{2} + 36 a + 1\right)\cdot 47^{2} + \left(15 a^{3} + 34 a^{2} + 7 a + 30\right)\cdot 47^{3} + \left(9 a^{3} + a^{2} + 41 a + 41\right)\cdot 47^{4} +O(47^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 32 a^{3} + 13 a^{2} + 30 a + 43 + \left(17 a^{3} + 45 a^{2} + 30 a + 29\right)\cdot 47 + \left(7 a^{3} + 14 a^{2} + 44 a + 17\right)\cdot 47^{2} + \left(16 a^{3} + 43 a^{2} + 14 a + 46\right)\cdot 47^{3} + \left(41 a^{3} + 40 a^{2} + 24 a\right)\cdot 47^{4} +O(47^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 30 a^{3} + 40 a^{2} + 18 a + 44 + \left(32 a^{3} + 19 a^{2} + 34 a\right)\cdot 47 + \left(7 a^{3} + 24 a^{2} + 22 a + 18\right)\cdot 47^{2} + \left(23 a^{3} + 29 a^{2} + 46 a + 13\right)\cdot 47^{3} + \left(28 a^{3} + 29 a^{2} + 27 a + 40\right)\cdot 47^{4} +O(47^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( a^{3} + 43 a^{2} + 45 a + 41 + \left(9 a^{3} + 43 a^{2} + 33 a + 8\right)\cdot 47 + \left(41 a^{3} + 5 a^{2} + 36 a + 12\right)\cdot 47^{2} + \left(a^{3} + 34 a^{2} + 3 a + 36\right)\cdot 47^{3} + \left(25 a^{3} + 41 a^{2} + 40 a + 39\right)\cdot 47^{4} +O(47^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 45 a^{3} + 13 a^{2} + 27 a + 19 + \left(36 a^{3} + 37 a^{2} + 25 a + 21\right)\cdot 47 + \left(22 a^{3} + 18 a^{2} + 9 a + 28\right)\cdot 47^{2} + \left(a^{3} + 30 a^{2} + 9 a + 9\right)\cdot 47^{3} + \left(20 a^{3} + 25 a^{2} + 11 a + 13\right)\cdot 47^{4} +O(47^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 30 a^{3} + 28 a^{2} + 11 a + 39 + \left(18 a^{3} + 38 a^{2} + 16 a + 34\right)\cdot 47 + \left(10 a^{3} + 15 a^{2} + 39 a + 17\right)\cdot 47^{2} + \left(41 a^{3} + 23 a^{2} + 6 a + 18\right)\cdot 47^{3} + \left(18 a^{3} + 31 a^{2} + 3 a + 24\right)\cdot 47^{4} +O(47^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 21 a^{3} + 30 a^{2} + 30 a + 12 + \left(9 a^{3} + 31 a^{2} + 38 a + 13\right)\cdot 47 + \left(12 a^{3} + 21 a^{2} + 33 a + 1\right)\cdot 47^{2} + \left(36 a^{3} + 39 a^{2} + 12 a + 28\right)\cdot 47^{3} + \left(30 a^{3} + 28 a^{2} + 43 a + 41\right)\cdot 47^{4} +O(47^{5})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 5 a^{3} + 36 a^{2} + 41 a + 30 + \left(37 a^{3} + 28 a^{2} + 19 a + 29\right)\cdot 47 + \left(40 a^{3} + 6 a^{2} + 45 a + 46\right)\cdot 47^{2} + \left(40 a^{3} + 7 a^{2} + 28 a + 31\right)\cdot 47^{3} + \left(13 a^{3} + 12 a^{2} + 28 a + 1\right)\cdot 47^{4} +O(47^{5})\)  Toggle raw display
$r_{ 9 }$ $=$ \( 27 a^{3} + 5 a^{2} + 5 a + 2 + \left(6 a^{3} + 9 a + 33\right)\cdot 47 + \left(38 a^{3} + a^{2} + 28 a + 38\right)\cdot 47^{2} + \left(13 a^{3} + 14 a^{2} + 3 a + 46\right)\cdot 47^{3} + \left(10 a^{3} + 11 a^{2} + 13 a + 31\right)\cdot 47^{4} +O(47^{5})\)  Toggle raw display
$r_{ 10 }$ $=$ \( 31 a^{3} + 7 a^{2} + 38 a + 45 + \left(6 a^{3} + 38 a^{2} + 22 a + 8\right)\cdot 47 + \left(15 a^{3} + 23 a^{2} + 21 a + 7\right)\cdot 47^{2} + \left(41 a^{3} + 18 a^{2} + 31 a + 29\right)\cdot 47^{3} + \left(3 a^{3} + 39 a^{2} + 28 a + 6\right)\cdot 47^{4} +O(47^{5})\)  Toggle raw display
$r_{ 11 }$ $=$ \( 17 a^{3} + 31 a^{2} + 31 a + 3 + \left(41 a^{3} + 21 a^{2} + 11 a + 44\right)\cdot 47 + \left(14 a^{3} + 45 a^{2} + 26 a + 38\right)\cdot 47^{2} + \left(2 a^{3} + 10 a^{2} + 2 a + 3\right)\cdot 47^{3} + \left(45 a^{3} + 34 a^{2} + 14 a + 46\right)\cdot 47^{4} +O(47^{5})\)  Toggle raw display
$r_{ 12 }$ $=$ \( 4 a^{3} + 31 a^{2} + 39 a + 23 + \left(13 a^{3} + 32 a^{2} + 40 a + 32\right)\cdot 47 + \left(24 a^{3} + 14 a^{2} + 30 a + 6\right)\cdot 47^{2} + \left(a^{3} + 43 a^{2} + 19 a + 35\right)\cdot 47^{3} + \left(35 a^{3} + 31 a^{2} + 6 a + 40\right)\cdot 47^{4} +O(47^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 12 }$ Character values
$c1$ $c2$ $c3$ $c4$
$1$ $1$ $()$ $1$ $1$ $1$ $1$
$1$ $2$ $(1,7)(2,8)(3,9)(4,10)(5,11)(6,12)$ $-1$ $-1$ $-1$ $-1$
$1$ $3$ $(1,3,5)(2,4,12)(6,8,10)(7,9,11)$ $\zeta_{12}^{2} - 1$ $\zeta_{12}^{2} - 1$ $-\zeta_{12}^{2}$ $-\zeta_{12}^{2}$
$1$ $3$ $(1,5,3)(2,12,4)(6,10,8)(7,11,9)$ $-\zeta_{12}^{2}$ $-\zeta_{12}^{2}$ $\zeta_{12}^{2} - 1$ $\zeta_{12}^{2} - 1$
$1$ $4$ $(1,6,7,12)(2,3,8,9)(4,5,10,11)$ $\zeta_{12}^{3}$ $-\zeta_{12}^{3}$ $\zeta_{12}^{3}$ $-\zeta_{12}^{3}$
$1$ $4$ $(1,12,7,6)(2,9,8,3)(4,11,10,5)$ $-\zeta_{12}^{3}$ $\zeta_{12}^{3}$ $-\zeta_{12}^{3}$ $\zeta_{12}^{3}$
$1$ $6$ $(1,11,3,7,5,9)(2,6,4,8,12,10)$ $\zeta_{12}^{2}$ $\zeta_{12}^{2}$ $-\zeta_{12}^{2} + 1$ $-\zeta_{12}^{2} + 1$
$1$ $6$ $(1,9,5,7,3,11)(2,10,12,8,4,6)$ $-\zeta_{12}^{2} + 1$ $-\zeta_{12}^{2} + 1$ $\zeta_{12}^{2}$ $\zeta_{12}^{2}$
$1$ $12$ $(1,2,11,6,3,4,7,8,5,12,9,10)$ $\zeta_{12}$ $-\zeta_{12}$ $\zeta_{12}^{3} - \zeta_{12}$ $-\zeta_{12}^{3} + \zeta_{12}$
$1$ $12$ $(1,4,9,6,5,2,7,10,3,12,11,8)$ $\zeta_{12}^{3} - \zeta_{12}$ $-\zeta_{12}^{3} + \zeta_{12}$ $\zeta_{12}$ $-\zeta_{12}$
$1$ $12$ $(1,8,11,12,3,10,7,2,5,6,9,4)$ $-\zeta_{12}$ $\zeta_{12}$ $-\zeta_{12}^{3} + \zeta_{12}$ $\zeta_{12}^{3} - \zeta_{12}$
$1$ $12$ $(1,10,9,12,5,8,7,4,3,6,11,2)$ $-\zeta_{12}^{3} + \zeta_{12}$ $\zeta_{12}^{3} - \zeta_{12}$ $-\zeta_{12}$ $\zeta_{12}$
The blue line marks the conjugacy class containing complex conjugation.