Basic invariants
| Dimension: | $45$ |
| Group: | $A_8$ |
| Conductor: | \(782\!\cdots\!528\)\(\medspace = 2^{100} \cdot 3217^{39} \) |
| Artin number field: | Galois closure of 8.0.72641749645773438449680384.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | 336 |
| Parity: | even |
| Projective image: | $A_8$ |
| Projective field: | Galois closure of 8.0.72641749645773438449680384.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 179 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 179 }$:
\( x^{2} + 172x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 16 + 153\cdot 179 + 69\cdot 179^{2} + 79\cdot 179^{3} + 94\cdot 179^{4} + 38\cdot 179^{5} + 161\cdot 179^{6} + 108\cdot 179^{7} +O(179^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 137 + 154\cdot 179 + 67\cdot 179^{2} + 54\cdot 179^{3} + 108\cdot 179^{4} + 156\cdot 179^{5} + 26\cdot 179^{6} + 172\cdot 179^{7} +O(179^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 122 + 83\cdot 179 + 163\cdot 179^{2} + 12\cdot 179^{3} + 177\cdot 179^{4} + 156\cdot 179^{5} + 26\cdot 179^{6} + 39\cdot 179^{7} +O(179^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 137 a + 138 + \left(176 a + 107\right)\cdot 179 + \left(7 a + 119\right)\cdot 179^{2} + \left(113 a + 141\right)\cdot 179^{3} + 70\cdot 179^{4} + \left(90 a + 109\right)\cdot 179^{5} + \left(88 a + 127\right)\cdot 179^{6} + \left(20 a + 113\right)\cdot 179^{7} +O(179^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 46 a + 71 + \left(31 a + 178\right)\cdot 179 + \left(54 a + 127\right)\cdot 179^{2} + \left(172 a + 112\right)\cdot 179^{3} + \left(102 a + 56\right)\cdot 179^{4} + \left(113 a + 176\right)\cdot 179^{5} + \left(78 a + 108\right)\cdot 179^{6} + \left(30 a + 100\right)\cdot 179^{7} +O(179^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 178 + 89\cdot 179 + 50\cdot 179^{2} + 94\cdot 179^{3} + 178\cdot 179^{4} + 81\cdot 179^{5} + 135\cdot 179^{6} + 135\cdot 179^{7} +O(179^{8})\)
|
| $r_{ 7 }$ | $=$ |
\( 42 a + 23 + \left(2 a + 134\right)\cdot 179 + \left(171 a + 177\right)\cdot 179^{2} + \left(65 a + 29\right)\cdot 179^{3} + \left(178 a + 141\right)\cdot 179^{4} + \left(88 a + 22\right)\cdot 179^{5} + \left(90 a + 120\right)\cdot 179^{6} + \left(158 a + 168\right)\cdot 179^{7} +O(179^{8})\)
|
| $r_{ 8 }$ | $=$ |
\( 133 a + 35 + \left(147 a + 172\right)\cdot 179 + \left(124 a + 117\right)\cdot 179^{2} + \left(6 a + 11\right)\cdot 179^{3} + \left(76 a + 68\right)\cdot 179^{4} + \left(65 a + 152\right)\cdot 179^{5} + \left(100 a + 8\right)\cdot 179^{6} + \left(148 a + 56\right)\cdot 179^{7} +O(179^{8})\)
|
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 values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $45$ | $45$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $-3$ | $-3$ |
| $210$ | $2$ | $(1,2)(3,4)$ | $-3$ | $-3$ |
| $112$ | $3$ | $(1,2,3)$ | $0$ | $0$ |
| $1120$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ | $0$ |
| $1260$ | $4$ | $(1,2,3,4)(5,6,7,8)$ | $1$ | $1$ |
| $2520$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ | $1$ |
| $1344$ | $5$ | $(1,2,3,4,5)$ | $0$ | $0$ |
| $1680$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $0$ | $0$ |
| $3360$ | $6$ | $(1,2,3,4,5,6)(7,8)$ | $0$ | $0$ |
| $2880$ | $7$ | $(1,2,3,4,5,6,7)$ | $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ | $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
| $2880$ | $7$ | $(1,3,4,5,6,7,2)$ | $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ | $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ |
| $1344$ | $15$ | $(1,2,3,4,5)(6,7,8)$ | $0$ | $0$ |
| $1344$ | $15$ | $(1,3,4,5,2)(6,7,8)$ | $0$ | $0$ |