# Properties

 Label 56.498...664.105.a Dimension $56$ Group $A_8$ Conductor $4.988\times 10^{267}$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $56$ Group: $A_8$ Conductor: $$498\!\cdots\!664$$$$\medspace = 2^{138} \cdot 51473^{48}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 8.0.19501894337558159417628591379185664.1 Galois orbit size: $1$ Smallest permutation container: 105 Parity: even Projective image: $A_8$ Projective field: 8.0.19501894337558159417628591379185664.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$: $$x^{2} + 70 x + 5$$
Roots:
 $r_{ 1 }$ $=$ $$59 + 21\cdot 73 + 39\cdot 73^{2} + 45\cdot 73^{3} + 46\cdot 73^{4} + 66\cdot 73^{5} + 70\cdot 73^{6} + 68\cdot 73^{7} + 59\cdot 73^{8} + 64\cdot 73^{9} +O(73^{10})$$ $r_{ 2 }$ $=$ $$3 + 48\cdot 73 + 66\cdot 73^{2} + 60\cdot 73^{3} + 34\cdot 73^{4} + 58\cdot 73^{5} + 71\cdot 73^{6} + 71\cdot 73^{7} + 31\cdot 73^{8} + 55\cdot 73^{9} +O(73^{10})$$ $r_{ 3 }$ $=$ $$4 a + 33 + \left(69 a + 45\right)\cdot 73 + \left(47 a + 27\right)\cdot 73^{2} + \left(27 a + 47\right)\cdot 73^{3} + \left(36 a + 21\right)\cdot 73^{4} + \left(8 a + 47\right)\cdot 73^{5} + \left(15 a + 30\right)\cdot 73^{6} + \left(a + 59\right)\cdot 73^{7} + \left(68 a + 64\right)\cdot 73^{8} + \left(42 a + 13\right)\cdot 73^{9} +O(73^{10})$$ $r_{ 4 }$ $=$ $$69 a + 45 + \left(3 a + 29\right)\cdot 73 + \left(25 a + 29\right)\cdot 73^{2} + \left(45 a + 9\right)\cdot 73^{3} + \left(36 a + 30\right)\cdot 73^{4} + \left(64 a + 36\right)\cdot 73^{5} + \left(57 a + 67\right)\cdot 73^{6} + \left(71 a + 47\right)\cdot 73^{7} + \left(4 a + 48\right)\cdot 73^{8} + \left(30 a + 1\right)\cdot 73^{9} +O(73^{10})$$ $r_{ 5 }$ $=$ $$51 + 58\cdot 73 + 48\cdot 73^{2} + 11\cdot 73^{4} + 10\cdot 73^{5} + 70\cdot 73^{6} + 71\cdot 73^{7} + 22\cdot 73^{8} + 15\cdot 73^{9} +O(73^{10})$$ $r_{ 6 }$ $=$ $$53 + 30\cdot 73 + 61\cdot 73^{2} + 60\cdot 73^{3} + 66\cdot 73^{4} + 73^{5} + 3\cdot 73^{6} + 38\cdot 73^{7} + 54\cdot 73^{8} +O(73^{10})$$ $r_{ 7 }$ $=$ $$49 a + 60 + \left(38 a + 31\right)\cdot 73 + \left(12 a + 46\right)\cdot 73^{2} + 26 a\cdot 73^{3} + \left(7 a + 6\right)\cdot 73^{4} + \left(18 a + 12\right)\cdot 73^{5} + \left(50 a + 32\right)\cdot 73^{6} + \left(15 a + 41\right)\cdot 73^{7} + \left(56 a + 37\right)\cdot 73^{8} + \left(51 a + 20\right)\cdot 73^{9} +O(73^{10})$$ $r_{ 8 }$ $=$ $$24 a + 61 + \left(34 a + 25\right)\cdot 73 + \left(60 a + 45\right)\cdot 73^{2} + \left(46 a + 66\right)\cdot 73^{3} + \left(65 a + 1\right)\cdot 73^{4} + \left(54 a + 59\right)\cdot 73^{5} + \left(22 a + 18\right)\cdot 73^{6} + \left(57 a + 38\right)\cdot 73^{7} + \left(16 a + 44\right)\cdot 73^{8} + \left(21 a + 46\right)\cdot 73^{9} +O(73^{10})$$

### 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 values $c1$ $1$ $1$ $()$ $56$ $105$ $2$ $(1,2)(3,4)(5,6)(7,8)$ $8$ $210$ $2$ $(1,2)(3,4)$ $0$ $112$ $3$ $(1,2,3)$ $-4$ $1120$ $3$ $(1,2,3)(4,5,6)$ $-1$ $1260$ $4$ $(1,2,3,4)(5,6,7,8)$ $0$ $2520$ $4$ $(1,2,3,4)(5,6)$ $0$ $1344$ $5$ $(1,2,3,4,5)$ $1$ $1680$ $6$ $(1,2,3)(4,5)(6,7)$ $0$ $3360$ $6$ $(1,2,3,4,5,6)(7,8)$ $-1$ $2880$ $7$ $(1,2,3,4,5,6,7)$ $0$ $2880$ $7$ $(1,3,4,5,6,7,2)$ $0$ $1344$ $15$ $(1,2,3,4,5)(6,7,8)$ $1$ $1344$ $15$ $(1,3,4,5,2)(6,7,8)$ $1$
The blue line marks the conjugacy class containing complex conjugation.