Basic invariants
Dimension: | $56$ |
Group: | $A_8$ |
Conductor: | \(118\!\cdots\!336\)\(\medspace = 2^{112} \cdot 3217^{48} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 8.0.72641749645773438449680384.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 105 |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $A_8$ |
Projective stem field: | Galois closure of 8.0.72641749645773438449680384.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 4x^{7} + 3217 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 179 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 179 }$: \( x^{2} + 172x + 2 \)
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 value |
$1$ | $1$ | $()$ | $56$ |
$105$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $8$ |
$210$ | $2$ | $(1,2)(3,4)$ | $0$ |
$112$ | $3$ | $(1,2,3)$ | $-4$ |
$1120$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
$1260$ | $4$ | $(1,2,3,4)(5,6,7,8)$ | $0$ |
$2520$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
$1344$ | $5$ | $(1,2,3,4,5)$ | $1$ |
$1680$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $0$ |
$3360$ | $6$ | $(1,2,3,4,5,6)(7,8)$ | $-1$ |
$2880$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
$2880$ | $7$ | $(1,3,4,5,6,7,2)$ | $0$ |
$1344$ | $15$ | $(1,2,3,4,5)(6,7,8)$ | $1$ |
$1344$ | $15$ | $(1,3,4,5,2)(6,7,8)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.