Basic invariants
Dimension: | $2$ |
Group: | $C_8:C_2$ |
Conductor: | \(11025\)\(\medspace = 3^{2} \cdot 5^{2} \cdot 7^{2} \) |
Artin stem field: | Galois closure of 8.4.136744453125.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_8:C_2$ |
Parity: | odd |
Determinant: | 1.5.4t1.a.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{5})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 2x^{7} - 2x^{6} + 31x^{5} - 65x^{4} - 239x^{3} + 58x^{2} + 868x + 961 \) . |
The roots of $f$ are computed in $\Q_{ 181 }$ to precision 8.
Roots:
$r_{ 1 }$ | $=$ |
\( 52 + 73\cdot 181 + 23\cdot 181^{2} + 118\cdot 181^{3} + 98\cdot 181^{4} + 43\cdot 181^{5} + 33\cdot 181^{6} + 101\cdot 181^{7} +O(181^{8})\)
$r_{ 2 }$ |
$=$ |
\( 58 + 2\cdot 181 + 113\cdot 181^{2} + 177\cdot 181^{3} + 170\cdot 181^{4} + 158\cdot 181^{5} + 102\cdot 181^{6} + 105\cdot 181^{7} +O(181^{8})\)
| $r_{ 3 }$ |
$=$ |
\( 69 + 95\cdot 181 + 116\cdot 181^{2} + 167\cdot 181^{3} + 77\cdot 181^{4} + 124\cdot 181^{5} + 95\cdot 181^{6} + 40\cdot 181^{7} +O(181^{8})\)
| $r_{ 4 }$ |
$=$ |
\( 80 + 178\cdot 181 + 45\cdot 181^{2} + 93\cdot 181^{3} + 127\cdot 181^{4} + 93\cdot 181^{5} + 25\cdot 181^{6} + 98\cdot 181^{7} +O(181^{8})\)
| $r_{ 5 }$ |
$=$ |
\( 90 + 174\cdot 181 + 47\cdot 181^{2} + 179\cdot 181^{3} + 63\cdot 181^{4} + 35\cdot 181^{5} + 86\cdot 181^{6} + 83\cdot 181^{7} +O(181^{8})\)
| $r_{ 6 }$ |
$=$ |
\( 121 + 72\cdot 181 + 135\cdot 181^{2} + 117\cdot 181^{3} + 96\cdot 181^{4} + 76\cdot 181^{5} + 77\cdot 181^{6} + 26\cdot 181^{7} +O(181^{8})\)
| $r_{ 7 }$ |
$=$ |
\( 124 + 94\cdot 181 + 151\cdot 181^{2} + 102\cdot 181^{3} + 92\cdot 181^{4} + 108\cdot 181^{5} + 154\cdot 181^{6} + 139\cdot 181^{7} +O(181^{8})\)
| $r_{ 8 }$ |
$=$ |
\( 132 + 32\cdot 181 + 90\cdot 181^{2} + 129\cdot 181^{3} + 176\cdot 181^{4} + 82\cdot 181^{5} + 148\cdot 181^{6} + 128\cdot 181^{7} +O(181^{8})\)
| |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,2)(3,5)(4,7)(6,8)$ | $-2$ |
$2$ | $2$ | $(1,2)(6,8)$ | $0$ |
$1$ | $4$ | $(1,8,2,6)(3,4,5,7)$ | $2 \zeta_{4}$ |
$1$ | $4$ | $(1,6,2,8)(3,7,5,4)$ | $-2 \zeta_{4}$ |
$2$ | $4$ | $(1,6,2,8)(3,4,5,7)$ | $0$ |
$2$ | $8$ | $(1,7,8,3,2,4,6,5)$ | $0$ |
$2$ | $8$ | $(1,3,6,7,2,5,8,4)$ | $0$ |
$2$ | $8$ | $(1,4,6,3,2,7,8,5)$ | $0$ |
$2$ | $8$ | $(1,3,8,4,2,5,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.