Properties

Label 2.18775.10t3.a.a
Dimension $2$
Group $D_{10}$
Conductor $18775$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{10}$
Conductor: \(18775\)\(\medspace = 5^{2} \cdot 751 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 10.0.746534197277346875.2
Galois orbit size: $2$
Smallest permutation container: $D_{10}$
Parity: odd
Determinant: 1.751.2t1.a.a
Projective image: $D_5$
Projective stem field: 5.1.564001.1

Defining polynomial

$f(x)$$=$\(x^{10} + 4 x^{8} - 68 x^{7} - 49 x^{6} - 262 x^{5} + 1050 x^{4} + 1550 x^{3} + 5925 x^{2} + 10849 x + 18989\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 9.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: \(x^{5} + 8 x + 57\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( a^{4} + 17 a^{3} + 28 a^{2} + 42 a + 30 + \left(47 a^{4} + 50 a^{3} + 18 a^{2} + 8 a + 29\right)\cdot 59 + \left(46 a^{4} + 47 a^{3} + 6 a^{2} + 7 a + 4\right)\cdot 59^{2} + \left(49 a^{4} + 6 a^{3} + 56 a\right)\cdot 59^{3} + \left(10 a^{4} + 4 a^{3} + 19 a^{2} + 22 a + 34\right)\cdot 59^{4} + \left(31 a^{4} + 23 a^{3} + 43 a^{2} + 38 a + 22\right)\cdot 59^{5} + \left(51 a^{4} + 9 a^{3} + a^{2} + 43 a + 58\right)\cdot 59^{6} + \left(52 a^{4} + 57 a^{3} + 16 a^{2} + 17 a + 7\right)\cdot 59^{7} + \left(7 a^{4} + 11 a^{3} + 4 a^{2} + 2 a + 15\right)\cdot 59^{8} +O(59^{9})\)  Toggle raw display
$r_{ 2 }$ $=$ \( a^{4} + 34 a^{3} + a^{2} + 50 a + 30 + \left(40 a^{4} + 50 a^{3} + 24 a^{2} + 20 a + 55\right)\cdot 59 + \left(45 a^{4} + 10 a^{3} + 40 a^{2} + 9 a + 20\right)\cdot 59^{2} + \left(46 a^{4} + 6 a^{3} + 51 a^{2} + 53 a + 4\right)\cdot 59^{3} + \left(39 a^{4} + 47 a^{3} + 26 a^{2} + 38 a + 54\right)\cdot 59^{4} + \left(28 a^{3} + 2 a^{2} + 9 a + 27\right)\cdot 59^{5} + \left(16 a^{4} + 28 a^{3} + 18 a^{2} + 12 a + 43\right)\cdot 59^{6} + \left(9 a^{4} + 41 a^{3} + 3 a^{2} + 50 a + 35\right)\cdot 59^{7} + \left(15 a^{4} + a^{3} + 2 a^{2} + 30 a + 2\right)\cdot 59^{8} +O(59^{9})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 19 a^{4} + 41 a^{3} + 33 a^{2} + 16 a + 39 + \left(54 a^{4} + 11 a^{3} + 40 a^{2} + 27 a + 52\right)\cdot 59 + \left(41 a^{4} + 54 a^{3} + 27 a^{2} + 44 a + 55\right)\cdot 59^{2} + \left(48 a^{4} + 42 a^{3} + 33 a^{2} + 17 a + 4\right)\cdot 59^{3} + \left(58 a^{4} + 34 a^{3} + 4 a^{2} + 46\right)\cdot 59^{4} + \left(46 a^{4} + 55 a^{3} + a^{2} + 6 a + 5\right)\cdot 59^{5} + \left(8 a^{4} + 41 a^{3} + 4 a^{2} + 2 a + 9\right)\cdot 59^{6} + \left(31 a^{4} + 3 a^{3} + 16 a^{2} + 48 a + 34\right)\cdot 59^{7} + \left(57 a^{4} + 36 a^{3} + 54 a^{2} + 5 a + 49\right)\cdot 59^{8} +O(59^{9})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 20 a^{4} + 20 a^{3} + 3 a^{2} + 16 a + 10 + \left(48 a^{4} + 42 a^{3} + 52 a^{2} + 32 a + 26\right)\cdot 59 + \left(57 a^{4} + 15 a^{3} + 19 a^{2} + 58 a + 51\right)\cdot 59^{2} + \left(46 a^{4} + 26 a^{3} + 48 a^{2} + 49 a + 52\right)\cdot 59^{3} + \left(41 a^{4} + 34 a^{3} + 29 a^{2} + 4 a + 7\right)\cdot 59^{4} + \left(22 a^{4} + 32 a^{3} + 7 a^{2} + 41 a + 39\right)\cdot 59^{5} + \left(38 a^{4} + 53 a^{3} + 36 a^{2} + 27 a + 9\right)\cdot 59^{6} + \left(35 a^{4} + 28 a^{3} + 41 a^{2} + 24 a + 51\right)\cdot 59^{7} + \left(25 a^{4} + 55 a^{3} + 47 a^{2} + 6 a + 45\right)\cdot 59^{8} +O(59^{9})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 22 a^{4} + 25 a^{3} + 34 a^{2} + 53 a + 11 + \left(35 a^{4} + 14 a^{3} + 44 a^{2} + 56 a + 2\right)\cdot 59 + \left(50 a^{4} + 40 a^{3} + 12 a^{2} + 57 a + 17\right)\cdot 59^{2} + \left(52 a^{4} + 24 a^{3} + 41 a^{2} + 19 a + 43\right)\cdot 59^{3} + \left(52 a^{4} + 6 a^{3} + 8 a^{2} + 50 a + 31\right)\cdot 59^{4} + \left(7 a^{4} + 41 a^{3} + 35 a^{2} + 30 a + 50\right)\cdot 59^{5} + \left(28 a^{4} + 57 a^{3} + 56 a^{2} + 2 a + 14\right)\cdot 59^{6} + \left(13 a^{4} + 37 a^{3} + 30 a^{2} + 40 a + 27\right)\cdot 59^{7} + \left(44 a^{4} + 16 a^{3} + 51 a^{2} + 48 a + 23\right)\cdot 59^{8} +O(59^{9})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 24 a^{4} + 54 a^{3} + 16 a^{2} + 48 a + 12 + \left(8 a^{4} + 4 a^{3} + 4 a^{2} + 24 a + 30\right)\cdot 59 + \left(35 a^{4} + 29 a^{3} + 44 a^{2} + 56 a + 12\right)\cdot 59^{2} + \left(52 a^{4} + 50 a^{3} + 37 a^{2} + 57 a + 6\right)\cdot 59^{3} + \left(7 a^{4} + 22 a^{3} + 55 a^{2} + 34 a + 15\right)\cdot 59^{4} + \left(38 a^{4} + 21 a^{3} + 34 a^{2} + 38 a + 55\right)\cdot 59^{5} + \left(38 a^{4} + 13 a^{3} + 48 a^{2} + 58 a + 34\right)\cdot 59^{6} + \left(23 a^{4} + 3 a^{3} + 21 a^{2} + 6 a + 21\right)\cdot 59^{7} + \left(28 a^{4} + 4 a^{3} + 58 a^{2} + a + 28\right)\cdot 59^{8} +O(59^{9})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 26 a^{4} + 38 a^{2} + 36 a + 13 + \left(46 a^{4} + 56 a^{3} + 15 a^{2} + 5 a + 2\right)\cdot 59 + \left(15 a^{4} + 13 a^{3} + 7 a^{2} + 24 a + 42\right)\cdot 59^{2} + \left(34 a^{4} + 51 a^{3} + 42 a^{2} + 26 a + 18\right)\cdot 59^{3} + \left(5 a^{4} + 27 a^{3} + a^{2} + 11 a\right)\cdot 59^{4} + \left(26 a^{4} + 23 a^{3} + 2 a^{2} + 14 a + 49\right)\cdot 59^{5} + \left(56 a^{4} + 17 a^{3} + 27 a^{2} + 25 a + 30\right)\cdot 59^{6} + \left(3 a^{4} + 56 a^{3} + 22 a^{2} + 32 a + 13\right)\cdot 59^{7} + \left(45 a^{4} + a^{3} + 52 a^{2} + 20 a + 5\right)\cdot 59^{8} +O(59^{9})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 28 a^{4} + 53 a^{3} + 24 a^{2} + 47 a + 14 + \left(41 a^{4} + 32 a^{3} + 24 a^{2} + 17 a + 41\right)\cdot 59 + \left(10 a^{4} + 25 a^{3} + 20 a^{2} + 39 a + 56\right)\cdot 59^{2} + \left(47 a^{4} + 44 a^{3} + 33 a^{2} + 29 a + 6\right)\cdot 59^{3} + \left(42 a^{4} + 8 a^{3} + 54 a^{2} + 9 a + 26\right)\cdot 59^{4} + \left(12 a^{4} + 5 a^{3} + 12 a^{2} + 9 a + 22\right)\cdot 59^{5} + \left(22 a^{4} + 21 a^{3} + 50 a^{2} + 42 a + 12\right)\cdot 59^{6} + \left(42 a^{4} + 31 a^{3} + 23 a^{2} + 5 a + 35\right)\cdot 59^{7} + \left(4 a^{4} + 39 a^{3} + 23 a^{2} + 43 a + 6\right)\cdot 59^{8} +O(59^{9})\)  Toggle raw display
$r_{ 9 }$ $=$ \( 45 a^{4} + 33 a^{3} + 47 a^{2} + 24 a + 52 + \left(31 a^{4} + 46 a^{3} + 18 a^{2} + 34 a + 49\right)\cdot 59 + \left(26 a^{4} + 58 a^{3} + 27 a^{2} + 15 a + 51\right)\cdot 59^{2} + \left(39 a^{4} + 48 a^{3} + 57 a^{2} + 42 a + 51\right)\cdot 59^{3} + \left(14 a^{4} + 47 a^{3} + 17 a^{2} + 45 a + 34\right)\cdot 59^{4} + \left(13 a^{4} + 35 a^{3} + 19 a^{2} + 49 a + 37\right)\cdot 59^{5} + \left(26 a^{4} + 20 a^{3} + 40 a^{2} + 4 a + 2\right)\cdot 59^{6} + \left(22 a^{4} + 56 a^{3} + 14 a^{2} + 4 a + 2\right)\cdot 59^{7} + \left(51 a^{4} + 6 a^{3} + 43 a^{2} + 6 a + 22\right)\cdot 59^{8} +O(59^{9})\)  Toggle raw display
$r_{ 10 }$ $=$ \( 50 a^{4} + 18 a^{3} + 12 a^{2} + 22 a + 25 + \left(44 a^{3} + 52 a^{2} + 7 a + 5\right)\cdot 59 + \left(23 a^{4} + 57 a^{3} + 29 a^{2} + 41 a + 41\right)\cdot 59^{2} + \left(53 a^{4} + 51 a^{3} + 8 a^{2} + 46\right)\cdot 59^{3} + \left(19 a^{4} + a^{3} + 17 a^{2} + 17 a + 44\right)\cdot 59^{4} + \left(36 a^{4} + 28 a^{3} + 18 a^{2} + 57 a + 43\right)\cdot 59^{5} + \left(8 a^{4} + 31 a^{3} + 12 a^{2} + 16 a + 19\right)\cdot 59^{6} + \left(a^{4} + 37 a^{3} + 45 a^{2} + 6 a + 7\right)\cdot 59^{7} + \left(15 a^{4} + 2 a^{3} + 16 a^{2} + 12 a + 37\right)\cdot 59^{8} +O(59^{9})\)  Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 10 }$

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.