Properties

Label 2.484.15t4.a.d
Dimension $2$
Group $S_3 \times C_5$
Conductor $484$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $S_3 \times C_5$
Conductor: \(484\)\(\medspace = 2^{2} \cdot 11^{2}\)
Artin stem field: 15.5.35351257235385344.1
Galois orbit size: $4$
Smallest permutation container: $S_3 \times C_5$
Parity: odd
Determinant: 1.11.10t1.a.d
Projective image: $S_3$
Projective stem field: 3.1.44.1

Defining polynomial

$f(x)$$=$\(x^{15} - 3 x^{13} - 5 x^{12} + 8 x^{11} + x^{10} + 14 x^{9} - 13 x^{8} - 17 x^{7} + 4 x^{6} + x^{5} + 22 x^{4} - 2 x^{3} - 8 x^{2} - 3 x - 1\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: \(x^{5} + x + 42\)  Toggle raw display

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.