Basic invariants
| Dimension: | $45$ |
| Group: | $A_8$ |
| Conductor: | \(193\!\cdots\!584\)\(\medspace = 2^{130} \cdot 52706761^{39} \) |
| Artin stem field: | Galois closure of 8.0.5620020392178081715128903113480438691650659827318784.1 |
| 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 8.0.5620020392178081715128903113480438691650659827318784.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 112x^{6} - 896x^{5} - 3360x^{4} - 7168x^{3} - 8960x^{2} - 6144x + 210825252 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 239 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 239 }$:
\( x^{2} + 237x + 7 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 107 a + 177 + \left(226 a + 157\right)\cdot 239 + \left(192 a + 95\right)\cdot 239^{2} + \left(110 a + 152\right)\cdot 239^{3} + \left(97 a + 158\right)\cdot 239^{4} + \left(209 a + 215\right)\cdot 239^{5} + \left(117 a + 49\right)\cdot 239^{6} + \left(52 a + 138\right)\cdot 239^{7} + \left(214 a + 132\right)\cdot 239^{8} + \left(170 a + 113\right)\cdot 239^{9} +O(239^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 175 + 239 + 225\cdot 239^{2} + 204\cdot 239^{3} + 136\cdot 239^{4} + 199\cdot 239^{5} + 175\cdot 239^{6} + 225\cdot 239^{7} + 83\cdot 239^{8} + 206\cdot 239^{9} +O(239^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 225 a + 228 + \left(189 a + 88\right)\cdot 239 + \left(177 a + 10\right)\cdot 239^{2} + \left(166 a + 117\right)\cdot 239^{3} + \left(76 a + 8\right)\cdot 239^{4} + \left(225 a + 52\right)\cdot 239^{5} + \left(5 a + 21\right)\cdot 239^{6} + \left(194 a + 2\right)\cdot 239^{7} + \left(133 a + 67\right)\cdot 239^{8} + \left(224 a + 138\right)\cdot 239^{9} +O(239^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 139 + 147\cdot 239 + 124\cdot 239^{2} + 79\cdot 239^{3} + 47\cdot 239^{4} + 4\cdot 239^{5} + 160\cdot 239^{6} + 36\cdot 239^{7} + 129\cdot 239^{8} + 124\cdot 239^{9} +O(239^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 132 a + 152 + \left(12 a + 25\right)\cdot 239 + \left(46 a + 16\right)\cdot 239^{2} + \left(128 a + 181\right)\cdot 239^{3} + \left(141 a + 3\right)\cdot 239^{4} + \left(29 a + 59\right)\cdot 239^{5} + \left(121 a + 76\right)\cdot 239^{6} + \left(186 a + 125\right)\cdot 239^{7} + \left(24 a + 30\right)\cdot 239^{8} + \left(68 a + 2\right)\cdot 239^{9} +O(239^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 14 a + 200 + \left(49 a + 4\right)\cdot 239 + \left(61 a + 176\right)\cdot 239^{2} + \left(72 a + 33\right)\cdot 239^{3} + \left(162 a + 234\right)\cdot 239^{4} + \left(13 a + 186\right)\cdot 239^{5} + \left(233 a + 46\right)\cdot 239^{6} + \left(44 a + 145\right)\cdot 239^{7} + \left(105 a + 140\right)\cdot 239^{8} + \left(14 a + 214\right)\cdot 239^{9} +O(239^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 68 + 4\cdot 239 + 82\cdot 239^{2} + 17\cdot 239^{3} + 234\cdot 239^{4} + 4\cdot 239^{5} + 183\cdot 239^{6} + 158\cdot 239^{7} + 200\cdot 239^{8} + 213\cdot 239^{9} +O(239^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 56 + 47\cdot 239 + 226\cdot 239^{2} + 169\cdot 239^{3} + 132\cdot 239^{4} + 233\cdot 239^{5} + 3\cdot 239^{6} + 124\cdot 239^{7} + 171\cdot 239^{8} + 181\cdot 239^{9} +O(239^{10})\)
|
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.