Basic invariants
| Dimension: | $45$ |
| Group: | $A_8$ |
| Conductor: | \(470\!\cdots\!888\)\(\medspace = 2^{130} \cdot 11^{39} \cdot 435593^{39} \) |
| Artin stem field: | Galois closure of 8.0.3172352108695376607450650959096645338500694016.1 |
| 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 8.0.3172352108695376607450650959096645338500694016.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 112x^{6} - 896x^{5} - 3360x^{4} - 7168x^{3} - 8960x^{2} - 6144x + 210825220 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$:
\( x^{2} + 97x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( a + 45 + \left(61 a + 21\right)\cdot 101 + \left(100 a + 77\right)\cdot 101^{2} + \left(57 a + 7\right)\cdot 101^{3} + \left(87 a + 82\right)\cdot 101^{4} + \left(92 a + 15\right)\cdot 101^{5} + \left(26 a + 55\right)\cdot 101^{6} + \left(58 a + 93\right)\cdot 101^{7} + \left(83 a + 93\right)\cdot 101^{8} + \left(65 a + 43\right)\cdot 101^{9} +O(101^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 a + 27 + \left(32 a + 74\right)\cdot 101 + \left(59 a + 81\right)\cdot 101^{2} + \left(9 a + 36\right)\cdot 101^{3} + \left(75 a + 34\right)\cdot 101^{4} + \left(77 a + 51\right)\cdot 101^{5} + \left(38 a + 82\right)\cdot 101^{6} + \left(96 a + 27\right)\cdot 101^{7} + \left(73 a + 32\right)\cdot 101^{8} + \left(69 a + 74\right)\cdot 101^{9} +O(101^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 74 a + 99 + \left(15 a + 1\right)\cdot 101 + \left(77 a + 10\right)\cdot 101^{2} + \left(99 a + 79\right)\cdot 101^{3} + \left(94 a + 98\right)\cdot 101^{4} + \left(59 a + 39\right)\cdot 101^{5} + \left(12 a + 42\right)\cdot 101^{6} + \left(77 a + 72\right)\cdot 101^{7} + \left(42 a + 12\right)\cdot 101^{8} + \left(a + 36\right)\cdot 101^{9} +O(101^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 100 a + 49 + \left(39 a + 62\right)\cdot 101 + 14\cdot 101^{2} + \left(43 a + 38\right)\cdot 101^{3} + \left(13 a + 71\right)\cdot 101^{4} + \left(8 a + 97\right)\cdot 101^{5} + \left(74 a + 69\right)\cdot 101^{6} + \left(42 a + 97\right)\cdot 101^{7} + \left(17 a + 66\right)\cdot 101^{8} + \left(35 a + 21\right)\cdot 101^{9} +O(101^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 89 a + 75 + \left(68 a + 89\right)\cdot 101 + \left(41 a + 84\right)\cdot 101^{2} + \left(91 a + 15\right)\cdot 101^{3} + \left(25 a + 22\right)\cdot 101^{4} + \left(23 a + 85\right)\cdot 101^{5} + \left(62 a + 58\right)\cdot 101^{6} + \left(4 a + 71\right)\cdot 101^{7} + \left(27 a + 29\right)\cdot 101^{8} + \left(31 a + 77\right)\cdot 101^{9} +O(101^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 27 a + 92 + \left(85 a + 91\right)\cdot 101 + \left(23 a + 100\right)\cdot 101^{2} + \left(a + 97\right)\cdot 101^{3} + \left(6 a + 75\right)\cdot 101^{4} + \left(41 a + 83\right)\cdot 101^{5} + \left(88 a + 32\right)\cdot 101^{6} + \left(23 a + 65\right)\cdot 101^{7} + \left(58 a + 5\right)\cdot 101^{8} + \left(99 a + 100\right)\cdot 101^{9} +O(101^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 65 a + 30 + \left(60 a + 43\right)\cdot 101 + \left(22 a + 2\right)\cdot 101^{2} + \left(29 a + 17\right)\cdot 101^{3} + \left(63 a + 49\right)\cdot 101^{4} + \left(10 a + 25\right)\cdot 101^{5} + \left(36 a + 65\right)\cdot 101^{6} + \left(2 a + 51\right)\cdot 101^{7} + \left(86 a + 11\right)\cdot 101^{8} + \left(53 a + 11\right)\cdot 101^{9} +O(101^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 36 a + 88 + \left(40 a + 18\right)\cdot 101 + \left(78 a + 32\right)\cdot 101^{2} + \left(71 a + 10\right)\cdot 101^{3} + \left(37 a + 71\right)\cdot 101^{4} + \left(90 a + 4\right)\cdot 101^{5} + \left(64 a + 98\right)\cdot 101^{6} + \left(98 a + 24\right)\cdot 101^{7} + \left(14 a + 50\right)\cdot 101^{8} + \left(47 a + 39\right)\cdot 101^{9} +O(101^{10})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 8 }$
| Cycle notation |
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.