# Properties

 Label 45.480...168.336.a.a Dimension $45$ Group $A_8$ Conductor $4.802\times 10^{221}$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $45$ Group: $A_8$ Conductor: $$480\!\cdots\!168$$$$\medspace = 2^{126} \cdot 51473^{39}$$ Artin stem field: Galois closure of 8.0.19501894337558159417628591379185664.1 Galois orbit size: $2$ Smallest permutation container: 336 Parity: even Determinant: 1.1.1t1.a.a Projective image: $A_8$ Projective stem field: Galois closure of 8.0.19501894337558159417628591379185664.1

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - 28x^{6} - 112x^{5} - 210x^{4} - 224x^{3} - 140x^{2} - 48x + 823561$$ x^8 - 28*x^6 - 112*x^5 - 210*x^4 - 224*x^3 - 140*x^2 - 48*x + 823561 .

The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$: $$x^{2} + 70x + 5$$

Roots:
 $r_{ 1 }$ $=$ $$59 + 21\cdot 73 + 39\cdot 73^{2} + 45\cdot 73^{3} + 46\cdot 73^{4} + 66\cdot 73^{5} + 70\cdot 73^{6} + 68\cdot 73^{7} + 59\cdot 73^{8} + 64\cdot 73^{9} +O(73^{10})$$ 59 + 21*73 + 39*73^2 + 45*73^3 + 46*73^4 + 66*73^5 + 70*73^6 + 68*73^7 + 59*73^8 + 64*73^9+O(73^10) $r_{ 2 }$ $=$ $$3 + 48\cdot 73 + 66\cdot 73^{2} + 60\cdot 73^{3} + 34\cdot 73^{4} + 58\cdot 73^{5} + 71\cdot 73^{6} + 71\cdot 73^{7} + 31\cdot 73^{8} + 55\cdot 73^{9} +O(73^{10})$$ 3 + 48*73 + 66*73^2 + 60*73^3 + 34*73^4 + 58*73^5 + 71*73^6 + 71*73^7 + 31*73^8 + 55*73^9+O(73^10) $r_{ 3 }$ $=$ $$4 a + 33 + \left(69 a + 45\right)\cdot 73 + \left(47 a + 27\right)\cdot 73^{2} + \left(27 a + 47\right)\cdot 73^{3} + \left(36 a + 21\right)\cdot 73^{4} + \left(8 a + 47\right)\cdot 73^{5} + \left(15 a + 30\right)\cdot 73^{6} + \left(a + 59\right)\cdot 73^{7} + \left(68 a + 64\right)\cdot 73^{8} + \left(42 a + 13\right)\cdot 73^{9} +O(73^{10})$$ 4*a + 33 + (69*a + 45)*73 + (47*a + 27)*73^2 + (27*a + 47)*73^3 + (36*a + 21)*73^4 + (8*a + 47)*73^5 + (15*a + 30)*73^6 + (a + 59)*73^7 + (68*a + 64)*73^8 + (42*a + 13)*73^9+O(73^10) $r_{ 4 }$ $=$ $$69 a + 45 + \left(3 a + 29\right)\cdot 73 + \left(25 a + 29\right)\cdot 73^{2} + \left(45 a + 9\right)\cdot 73^{3} + \left(36 a + 30\right)\cdot 73^{4} + \left(64 a + 36\right)\cdot 73^{5} + \left(57 a + 67\right)\cdot 73^{6} + \left(71 a + 47\right)\cdot 73^{7} + \left(4 a + 48\right)\cdot 73^{8} + \left(30 a + 1\right)\cdot 73^{9} +O(73^{10})$$ 69*a + 45 + (3*a + 29)*73 + (25*a + 29)*73^2 + (45*a + 9)*73^3 + (36*a + 30)*73^4 + (64*a + 36)*73^5 + (57*a + 67)*73^6 + (71*a + 47)*73^7 + (4*a + 48)*73^8 + (30*a + 1)*73^9+O(73^10) $r_{ 5 }$ $=$ $$51 + 58\cdot 73 + 48\cdot 73^{2} + 11\cdot 73^{4} + 10\cdot 73^{5} + 70\cdot 73^{6} + 71\cdot 73^{7} + 22\cdot 73^{8} + 15\cdot 73^{9} +O(73^{10})$$ 51 + 58*73 + 48*73^2 + 11*73^4 + 10*73^5 + 70*73^6 + 71*73^7 + 22*73^8 + 15*73^9+O(73^10) $r_{ 6 }$ $=$ $$53 + 30\cdot 73 + 61\cdot 73^{2} + 60\cdot 73^{3} + 66\cdot 73^{4} + 73^{5} + 3\cdot 73^{6} + 38\cdot 73^{7} + 54\cdot 73^{8} +O(73^{10})$$ 53 + 30*73 + 61*73^2 + 60*73^3 + 66*73^4 + 73^5 + 3*73^6 + 38*73^7 + 54*73^8+O(73^10) $r_{ 7 }$ $=$ $$49 a + 60 + \left(38 a + 31\right)\cdot 73 + \left(12 a + 46\right)\cdot 73^{2} + 26 a\cdot 73^{3} + \left(7 a + 6\right)\cdot 73^{4} + \left(18 a + 12\right)\cdot 73^{5} + \left(50 a + 32\right)\cdot 73^{6} + \left(15 a + 41\right)\cdot 73^{7} + \left(56 a + 37\right)\cdot 73^{8} + \left(51 a + 20\right)\cdot 73^{9} +O(73^{10})$$ 49*a + 60 + (38*a + 31)*73 + (12*a + 46)*73^2 + 26*a*73^3 + (7*a + 6)*73^4 + (18*a + 12)*73^5 + (50*a + 32)*73^6 + (15*a + 41)*73^7 + (56*a + 37)*73^8 + (51*a + 20)*73^9+O(73^10) $r_{ 8 }$ $=$ $$24 a + 61 + \left(34 a + 25\right)\cdot 73 + \left(60 a + 45\right)\cdot 73^{2} + \left(46 a + 66\right)\cdot 73^{3} + \left(65 a + 1\right)\cdot 73^{4} + \left(54 a + 59\right)\cdot 73^{5} + \left(22 a + 18\right)\cdot 73^{6} + \left(57 a + 38\right)\cdot 73^{7} + \left(16 a + 44\right)\cdot 73^{8} + \left(21 a + 46\right)\cdot 73^{9} +O(73^{10})$$ 24*a + 61 + (34*a + 25)*73 + (60*a + 45)*73^2 + (46*a + 66)*73^3 + (65*a + 1)*73^4 + (54*a + 59)*73^5 + (22*a + 18)*73^6 + (57*a + 38)*73^7 + (16*a + 44)*73^8 + (21*a + 46)*73^9+O(73^10)

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character value $1$ $1$ $()$ $45$ $105$ $2$ $(1,2)(3,4)(5,6)(7,8)$ $-3$ $210$ $2$ $(1,2)(3,4)$ $-3$ $112$ $3$ $(1,2,3)$ $0$ $1120$ $3$ $(1,2,3)(4,5,6)$ $0$ $1260$ $4$ $(1,2,3,4)(5,6,7,8)$ $1$ $2520$ $4$ $(1,2,3,4)(5,6)$ $1$ $1344$ $5$ $(1,2,3,4,5)$ $0$ $1680$ $6$ $(1,2,3)(4,5)(6,7)$ $0$ $3360$ $6$ $(1,2,3,4,5,6)(7,8)$ $0$ $2880$ $7$ $(1,2,3,4,5,6,7)$ $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ $2880$ $7$ $(1,3,4,5,6,7,2)$ $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ $1344$ $15$ $(1,2,3,4,5)(6,7,8)$ $0$ $1344$ $15$ $(1,3,4,5,2)(6,7,8)$ $0$

The blue line marks the conjugacy class containing complex conjugation.