# Properties

 Label 4.4811.8t44.b.a Dimension $4$ Group $C_2 \wr S_4$ Conductor $4811$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_2 \wr S_4$ Conductor: $$4811$$$$\medspace = 17 \cdot 283$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 8.0.1361513.1 Galois orbit size: $1$ Smallest permutation container: $C_2 \wr S_4$ Parity: odd Determinant: 1.4811.2t1.a.a Projective image: $C_2^3:S_4$ Projective stem field: Galois closure of 8.4.6689113369.1

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - x^{7} + 2x^{6} - 3x^{5} + 3x^{4} - 3x^{3} + 2x^{2} - x + 1$$ x^8 - x^7 + 2*x^6 - 3*x^5 + 3*x^4 - 3*x^3 + 2*x^2 - x + 1 .

The roots of $f$ are computed in an extension of $\Q_{ 79 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 79 }$: $$x^{2} + 78x + 3$$

Roots:
 $r_{ 1 }$ $=$ $$54 a + 44 + \left(39 a + 5\right)\cdot 79 + \left(47 a + 68\right)\cdot 79^{2} + \left(27 a + 42\right)\cdot 79^{3} + \left(47 a + 59\right)\cdot 79^{4} + \left(61 a + 6\right)\cdot 79^{5} + \left(31 a + 18\right)\cdot 79^{6} + \left(72 a + 49\right)\cdot 79^{7} + \left(57 a + 76\right)\cdot 79^{8} + \left(36 a + 67\right)\cdot 79^{9} +O(79^{10})$$ 54*a + 44 + (39*a + 5)*79 + (47*a + 68)*79^2 + (27*a + 42)*79^3 + (47*a + 59)*79^4 + (61*a + 6)*79^5 + (31*a + 18)*79^6 + (72*a + 49)*79^7 + (57*a + 76)*79^8 + (36*a + 67)*79^9+O(79^10) $r_{ 2 }$ $=$ $$60 a + 72 + \left(15 a + 11\right)\cdot 79 + \left(53 a + 40\right)\cdot 79^{2} + \left(25 a + 39\right)\cdot 79^{3} + \left(57 a + 25\right)\cdot 79^{4} + \left(73 a + 11\right)\cdot 79^{5} + \left(62 a + 51\right)\cdot 79^{6} + \left(24 a + 1\right)\cdot 79^{7} + \left(45 a + 35\right)\cdot 79^{8} + \left(76 a + 71\right)\cdot 79^{9} +O(79^{10})$$ 60*a + 72 + (15*a + 11)*79 + (53*a + 40)*79^2 + (25*a + 39)*79^3 + (57*a + 25)*79^4 + (73*a + 11)*79^5 + (62*a + 51)*79^6 + (24*a + 1)*79^7 + (45*a + 35)*79^8 + (76*a + 71)*79^9+O(79^10) $r_{ 3 }$ $=$ $$72 + 6\cdot 79 + 46\cdot 79^{2} + 26\cdot 79^{3} + 55\cdot 79^{4} + 62\cdot 79^{5} + 35\cdot 79^{6} + 36\cdot 79^{7} + 19\cdot 79^{8} + 68\cdot 79^{9} +O(79^{10})$$ 72 + 6*79 + 46*79^2 + 26*79^3 + 55*79^4 + 62*79^5 + 35*79^6 + 36*79^7 + 19*79^8 + 68*79^9+O(79^10) $r_{ 4 }$ $=$ $$25 a + 19 + \left(39 a + 70\right)\cdot 79 + \left(31 a + 75\right)\cdot 79^{2} + \left(51 a + 22\right)\cdot 79^{3} + 31 a\cdot 79^{4} + \left(17 a + 21\right)\cdot 79^{5} + \left(47 a + 67\right)\cdot 79^{6} + \left(6 a + 10\right)\cdot 79^{7} + \left(21 a + 62\right)\cdot 79^{8} + \left(42 a + 46\right)\cdot 79^{9} +O(79^{10})$$ 25*a + 19 + (39*a + 70)*79 + (31*a + 75)*79^2 + (51*a + 22)*79^3 + 31*a*79^4 + (17*a + 21)*79^5 + (47*a + 67)*79^6 + (6*a + 10)*79^7 + (21*a + 62)*79^8 + (42*a + 46)*79^9+O(79^10) $r_{ 5 }$ $=$ $$78 a + 46 + \left(28 a + 38\right)\cdot 79 + \left(40 a + 42\right)\cdot 79^{2} + \left(74 a + 27\right)\cdot 79^{3} + \left(32 a + 48\right)\cdot 79^{4} + \left(48 a + 33\right)\cdot 79^{5} + \left(52 a + 56\right)\cdot 79^{6} + \left(53 a + 19\right)\cdot 79^{7} + \left(21 a + 57\right)\cdot 79^{8} + \left(56 a + 69\right)\cdot 79^{9} +O(79^{10})$$ 78*a + 46 + (28*a + 38)*79 + (40*a + 42)*79^2 + (74*a + 27)*79^3 + (32*a + 48)*79^4 + (48*a + 33)*79^5 + (52*a + 56)*79^6 + (53*a + 19)*79^7 + (21*a + 57)*79^8 + (56*a + 69)*79^9+O(79^10) $r_{ 6 }$ $=$ $$45 + 67\cdot 79 + 69\cdot 79^{2} + 3\cdot 79^{3} + 63\cdot 79^{4} + 24\cdot 79^{5} + 65\cdot 79^{6} + 55\cdot 79^{7} + 63\cdot 79^{8} + 21\cdot 79^{9} +O(79^{10})$$ 45 + 67*79 + 69*79^2 + 3*79^3 + 63*79^4 + 24*79^5 + 65*79^6 + 55*79^7 + 63*79^8 + 21*79^9+O(79^10) $r_{ 7 }$ $=$ $$19 a + 53 + \left(63 a + 46\right)\cdot 79 + \left(25 a + 77\right)\cdot 79^{2} + \left(53 a + 11\right)\cdot 79^{3} + \left(21 a + 57\right)\cdot 79^{4} + \left(5 a + 27\right)\cdot 79^{5} + \left(16 a + 40\right)\cdot 79^{6} + \left(54 a + 42\right)\cdot 79^{7} + \left(33 a + 55\right)\cdot 79^{8} + \left(2 a + 23\right)\cdot 79^{9} +O(79^{10})$$ 19*a + 53 + (63*a + 46)*79 + (25*a + 77)*79^2 + (53*a + 11)*79^3 + (21*a + 57)*79^4 + (5*a + 27)*79^5 + (16*a + 40)*79^6 + (54*a + 42)*79^7 + (33*a + 55)*79^8 + (2*a + 23)*79^9+O(79^10) $r_{ 8 }$ $=$ $$a + 45 + \left(50 a + 68\right)\cdot 79 + \left(38 a + 53\right)\cdot 79^{2} + \left(4 a + 61\right)\cdot 79^{3} + \left(46 a + 6\right)\cdot 79^{4} + \left(30 a + 49\right)\cdot 79^{5} + \left(26 a + 60\right)\cdot 79^{6} + \left(25 a + 20\right)\cdot 79^{7} + \left(57 a + 25\right)\cdot 79^{8} + \left(22 a + 25\right)\cdot 79^{9} +O(79^{10})$$ a + 45 + (50*a + 68)*79 + (38*a + 53)*79^2 + (4*a + 61)*79^3 + (46*a + 6)*79^4 + (30*a + 49)*79^5 + (26*a + 60)*79^6 + (25*a + 20)*79^7 + (57*a + 25)*79^8 + (22*a + 25)*79^9+O(79^10)

## Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character value $1$ $1$ $()$ $4$ $1$ $2$ $(1,8)(2,7)(3,6)(4,5)$ $-4$ $4$ $2$ $(1,8)$ $2$ $4$ $2$ $(1,8)(2,7)(4,5)$ $-2$ $6$ $2$ $(1,8)(3,6)$ $0$ $12$ $2$ $(1,3)(2,4)(5,7)(6,8)$ $0$ $12$ $2$ $(3,4)(5,6)$ $2$ $12$ $2$ $(1,2)(3,6)(4,5)(7,8)$ $-2$ $24$ $2$ $(1,8)(3,4)(5,6)$ $0$ $32$ $3$ $(1,4,2)(5,7,8)$ $1$ $12$ $4$ $(1,6,8,3)(2,5,7,4)$ $0$ $12$ $4$ $(3,5,6,4)$ $2$ $12$ $4$ $(1,7,8,2)(3,6)(4,5)$ $-2$ $24$ $4$ $(1,6,8,3)(2,4)(5,7)$ $0$ $24$ $4$ $(1,8)(3,5,6,4)$ $0$ $48$ $4$ $(1,4,3,2)(5,6,7,8)$ $0$ $32$ $6$ $(1,5,7,8,4,2)$ $1$ $32$ $6$ $(1,4,2)(3,6)(5,7,8)$ $-1$ $32$ $6$ $(1,5,7,8,4,2)(3,6)$ $-1$ $48$ $8$ $(1,5,6,7,8,4,3,2)$ $0$

The blue line marks the conjugacy class containing complex conjugation.