Properties

Label 2.524.15t2.a.c
Dimension $2$
Group $D_{15}$
Conductor $524$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{15}$
Conductor: \(524\)\(\medspace = 2^{2} \cdot 131 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 15.1.677952124826430464.1
Galois orbit size: $4$
Smallest permutation container: $D_{15}$
Parity: odd
Determinant: 1.131.2t1.a.a
Projective image: $D_{15}$
Projective stem field: 15.1.677952124826430464.1

Defining polynomial

$f(x)$$=$\(x^{15} - x^{14} + 4 x^{13} + 4 x^{11} - 2 x^{10} - 6 x^{9} - 12 x^{8} - 25 x^{7} + 19 x^{6} - 6 x^{5} + 28 x^{4} + 32 x^{3} - 36 x^{2} + 4 x + 4\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 113 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 113 }$: \(x^{5} + 7 x + 110\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 5 a^{4} + 91 a^{3} + 17 a^{2} + 3 a + 54 + \left(90 a^{3} + 98 a^{2} + 11 a + 33\right)\cdot 113 + \left(86 a^{4} + 29 a^{3} + 25 a^{2} + 103 a + 7\right)\cdot 113^{2} + \left(105 a^{4} + 38 a^{3} + 91 a^{2} + 50 a + 90\right)\cdot 113^{3} + \left(88 a^{4} + 18 a^{3} + 70 a^{2} + 41 a + 63\right)\cdot 113^{4} +O(113^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 10 a^{4} + 92 a^{3} + 110 a^{2} + 20 a + 80 + \left(76 a^{4} + 60 a^{3} + 84 a^{2} + 58 a + 18\right)\cdot 113 + \left(18 a^{4} + 69 a^{3} + 84 a^{2} + 10 a + 78\right)\cdot 113^{2} + \left(52 a^{4} + 31 a^{3} + 86 a^{2} + 87 a + 87\right)\cdot 113^{3} + \left(25 a^{4} + 40 a^{3} + 23 a^{2} + 38 a + 106\right)\cdot 113^{4} +O(113^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 15 a^{4} + 77 a^{3} + 29 a^{2} + a + 102 + \left(105 a^{4} + 60 a^{3} + 61 a^{2} + 53 a + 58\right)\cdot 113 + \left(11 a^{4} + 99 a^{3} + 42 a^{2} + 2\right)\cdot 113^{2} + \left(12 a^{4} + 65 a^{3} + 100 a^{2} + 17 a + 6\right)\cdot 113^{3} + \left(58 a^{4} + 50 a^{3} + 93 a^{2} + 85 a + 27\right)\cdot 113^{4} +O(113^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 47 a^{4} + 73 a^{3} + 19 a^{2} + 43 a + 16 + \left(36 a^{4} + 30 a^{3} + 56 a^{2} + 8 a + 45\right)\cdot 113 + \left(48 a^{4} + 47 a^{3} + 103 a^{2} + 91 a + 63\right)\cdot 113^{2} + \left(62 a^{4} + 100 a^{3} + 10 a^{2} + 93 a + 9\right)\cdot 113^{3} + \left(24 a^{4} + 60 a^{3} + 66 a^{2} + 20 a + 79\right)\cdot 113^{4} +O(113^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 53 a^{4} + 33 a^{3} + 43 a^{2} + 74 a + 29 + \left(80 a^{4} + 17 a^{3} + 57 a^{2} + 45 a + 9\right)\cdot 113 + \left(93 a^{4} + 88 a^{3} + 101 a^{2} + 88 a + 73\right)\cdot 113^{2} + \left(89 a^{4} + 30 a^{3} + 88 a^{2} + 16 a + 68\right)\cdot 113^{3} + \left(43 a^{4} + 112 a^{3} + 22 a^{2} + 41 a + 59\right)\cdot 113^{4} +O(113^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 62 a^{4} + 85 a^{3} + 101 a^{2} + 80 a + 94 + \left(59 a^{4} + 34 a^{3} + 89 a^{2} + 9 a + 74\right)\cdot 113 + \left(47 a^{4} + 32 a^{3} + 60 a^{2} + 58 a + 43\right)\cdot 113^{2} + \left(38 a^{4} + 105 a^{3} + 109 a^{2} + 69 a + 40\right)\cdot 113^{3} + \left(83 a^{4} + 29 a^{3} + 4 a^{2} + 91 a + 55\right)\cdot 113^{4} +O(113^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 66 a^{4} + 19 a^{3} + 46 a^{2} + 112 a + 26 + \left(19 a^{4} + 7 a^{3} + 84 a^{2} + 24 a + 9\right)\cdot 113 + \left(29 a^{4} + 27 a^{3} + 45 a^{2} + 15 a + 99\right)\cdot 113^{2} + \left(69 a^{4} + 41 a^{3} + 32 a^{2} + 5 a + 54\right)\cdot 113^{3} + \left(86 a^{4} + 37 a^{3} + 16 a^{2} + 11 a + 28\right)\cdot 113^{4} +O(113^{5})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 69 a^{4} + 35 a^{3} + 83 a^{2} + 112 a + 96 + \left(20 a^{4} + 47 a^{3} + 40 a^{2} + 53 a + 80\right)\cdot 113 + \left(46 a^{4} + 63 a^{3} + 94 a^{2} + 25 a + 100\right)\cdot 113^{2} + \left(66 a^{4} + 50 a^{3} + 50 a^{2} + 94 a + 27\right)\cdot 113^{3} + \left(106 a^{4} + 55 a^{3} + 96 a^{2} + 25 a + 72\right)\cdot 113^{4} +O(113^{5})\)  Toggle raw display
$r_{ 9 }$ $=$ \( 79 a^{4} + 86 a^{3} + 14 a^{2} + 19 a + 37 + \left(47 a^{4} + 11 a^{3} + 75 a^{2} + 32 a + 108\right)\cdot 113 + \left(82 a^{4} + 52 a^{3} + 15 a^{2} + 5 a + 50\right)\cdot 113^{2} + \left(31 a^{4} + 11 a^{3} + 33 a^{2} + 71 a + 18\right)\cdot 113^{3} + \left(105 a^{4} + 93 a^{3} + 41 a^{2} + 99 a + 79\right)\cdot 113^{4} +O(113^{5})\)  Toggle raw display
$r_{ 10 }$ $=$ \( 94 a^{4} + 82 a^{3} + 11 a^{2} + 2 + \left(80 a^{4} + 94 a^{3} + 98 a^{2} + 60 a + 81\right)\cdot 113 + \left(50 a^{4} + 18 a^{3} + 78 a^{2} + 47 a + 106\right)\cdot 113^{2} + \left(80 a^{4} + 9 a^{3} + 59 a + 94\right)\cdot 113^{3} + \left(111 a^{4} + 50 a^{3} + 105 a^{2} + 59 a + 10\right)\cdot 113^{4} +O(113^{5})\)  Toggle raw display
$r_{ 11 }$ $=$ \( 95 a^{4} + 67 a^{3} + 9 a^{2} + 91 a + 104 + \left(44 a^{4} + 41 a^{3} + 99 a^{2} + 89 a + 1\right)\cdot 113 + \left(73 a^{4} + 25 a^{3} + 11 a^{2} + 12 a + 23\right)\cdot 113^{2} + \left(57 a^{4} + 16 a^{3} + 14 a^{2} + 25 a + 73\right)\cdot 113^{3} + \left(24 a^{4} + 61 a^{3} + 88 a^{2} + 21 a + 33\right)\cdot 113^{4} +O(113^{5})\)  Toggle raw display
$r_{ 12 }$ $=$ \( 100 a^{4} + 33 a^{3} + 44 a^{2} + 101 a + 21 + \left(50 a^{4} + 19 a^{3} + 105 a^{2} + 97 a + 92\right)\cdot 113 + \left(38 a^{4} + 75 a^{3} + 42 a^{2} + 15 a + 34\right)\cdot 113^{2} + \left(91 a^{4} + 58 a^{3} + 20 a^{2} + 107 a + 9\right)\cdot 113^{3} + \left(57 a^{4} + 38 a^{3} + 46 a^{2} + 45 a + 25\right)\cdot 113^{4} +O(113^{5})\)  Toggle raw display
$r_{ 13 }$ $=$ \( 102 a^{4} + 76 a^{3} + 39 a^{2} + 33 a + 92 + \left(73 a^{4} + 28 a^{3} + 5 a^{2} + 78 a + 64\right)\cdot 113 + \left(86 a^{4} + 48 a^{3} + 111 a^{2} + 104 a + 104\right)\cdot 113^{2} + \left(25 a^{4} + 4 a^{3} + 95 a^{2} + 74 a + 59\right)\cdot 113^{3} + \left(112 a^{4} + 58 a^{3} + 5 a^{2} + 91 a + 81\right)\cdot 113^{4} +O(113^{5})\)  Toggle raw display
$r_{ 14 }$ $=$ \( 108 a^{4} + 21 a^{3} + 74 a^{2} + 53 a + 109 + \left(20 a^{4} + 81 a^{3} + 23 a^{2} + 37 a + 48\right)\cdot 113 + \left(3 a^{4} + 31 a^{3} + 10 a^{2} + 106 a + 104\right)\cdot 113^{2} + \left(22 a^{4} + 66 a^{3} + 81 a^{2} + 61 a + 31\right)\cdot 113^{3} + \left(46 a^{4} + 83 a^{3} + 6 a^{2} + 45 a + 87\right)\cdot 113^{4} +O(113^{5})\)  Toggle raw display
$r_{ 15 }$ $=$ \( 112 a^{4} + 34 a^{3} + 39 a^{2} + 49 a + 43 + \left(73 a^{4} + 51 a^{3} + 37 a^{2} + 17 a + 63\right)\cdot 113 + \left(74 a^{4} + 82 a^{3} + 74 a^{2} + 106 a + 11\right)\cdot 113^{2} + \left(98 a^{4} + 47 a^{3} + 87 a^{2} + 69 a + 5\right)\cdot 113^{3} + \left(41 a^{4} + a^{3} + 102 a^{2} + 71 a + 94\right)\cdot 113^{4} +O(113^{5})\)  Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 15 }$

Cycle notation
$(1,6)(3,15)(4,11)(5,10)(7,12)(8,13)(9,14)$
$(1,9,10)(2,13,8)(3,12,11)(4,7,15)(5,14,6)$
$(1,15,8,12,5)(2,11,14,9,4)(3,6,10,7,13)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 15 }$ Character value
$1$$1$$()$$2$
$15$$2$$(1,6)(3,15)(4,11)(5,10)(7,12)(8,13)(9,14)$$0$
$2$$3$$(1,9,10)(2,13,8)(3,12,11)(4,7,15)(5,14,6)$$-1$
$2$$5$$(1,15,8,12,5)(2,11,14,9,4)(3,6,10,7,13)$$\zeta_{15}^{7} - \zeta_{15}^{3} + \zeta_{15}^{2} - 1$
$2$$5$$(1,8,5,15,12)(2,14,4,11,9)(3,10,13,6,7)$$-\zeta_{15}^{7} + \zeta_{15}^{3} - \zeta_{15}^{2}$
$2$$15$$(1,4,13,12,14,10,15,2,3,5,9,7,8,11,6)$$-\zeta_{15}^{7} + \zeta_{15}^{6} - \zeta_{15}^{4} + \zeta_{15}^{3} - \zeta_{15}^{2} + \zeta_{15} + 1$
$2$$15$$(1,13,14,15,3,9,8,6,4,12,10,2,5,7,11)$$-\zeta_{15}^{7} + \zeta_{15}^{5} - \zeta_{15}^{4} + \zeta_{15}^{2} - \zeta_{15} + 1$
$2$$15$$(1,14,3,8,4,10,5,11,13,15,9,6,12,2,7)$$-\zeta_{15}^{6} + \zeta_{15}^{4} - \zeta_{15}$
$2$$15$$(1,2,6,15,11,10,8,14,7,12,9,13,5,4,3)$$2 \zeta_{15}^{7} - \zeta_{15}^{5} + \zeta_{15}^{4} - \zeta_{15}^{3} + \zeta_{15} - 1$

The blue line marks the conjugacy class containing complex conjugation.