# Properties

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

# Related objects

## 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.a Projective image: $S_3$ Projective stem field: 3.1.44.1

## Defining polynomial

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

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

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

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

## 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

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

The blue line marks the conjugacy class containing complex conjugation.