Basic invariants
Dimension: | $2$ |
Group: | $C_8:C_2$ |
Conductor: | \(3757\)\(\medspace = 13 \cdot 17^{2} \) |
Artin stem field: | Galois closure of 8.0.69347235737.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_8:C_2$ |
Parity: | even |
Determinant: | 1.221.4t1.a.b |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{13}, \sqrt{17})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - x^{7} + 10x^{6} - 11x^{5} + 15x^{4} + 7x^{3} + 24x^{2} + 21x + 103 \) . |
The roots of $f$ are computed in $\Q_{ 101 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 6 + 48\cdot 101 + 11\cdot 101^{2} + 56\cdot 101^{3} + 88\cdot 101^{4} +O(101^{5})\)
$r_{ 2 }$ |
$=$ |
\( 47 + 85\cdot 101 + 71\cdot 101^{2} + 10\cdot 101^{3} + 41\cdot 101^{4} +O(101^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 51 + 97\cdot 101 + 62\cdot 101^{2} + 56\cdot 101^{3} + 10\cdot 101^{4} +O(101^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 62 + 30\cdot 101 + 60\cdot 101^{2} + 14\cdot 101^{3} + 66\cdot 101^{4} +O(101^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 79 + 38\cdot 101 + 77\cdot 101^{2} + 10\cdot 101^{3} + 92\cdot 101^{4} +O(101^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 80 + 63\cdot 101 + 76\cdot 101^{2} + 15\cdot 101^{3} + 35\cdot 101^{4} +O(101^{5})\)
| $r_{ 7 }$ |
$=$ |
\( 82 + 18\cdot 101 + 37\cdot 101^{2} + 7\cdot 101^{3} + 12\cdot 101^{4} +O(101^{5})\)
| $r_{ 8 }$ |
$=$ |
\( 99 + 20\cdot 101 + 6\cdot 101^{2} + 30\cdot 101^{3} + 58\cdot 101^{4} +O(101^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,8)(2,5)(3,7)(4,6)$ | $-2$ |
$2$ | $2$ | $(3,7)(4,6)$ | $0$ |
$1$ | $4$ | $(1,2,8,5)(3,4,7,6)$ | $2 \zeta_{4}$ |
$1$ | $4$ | $(1,5,8,2)(3,6,7,4)$ | $-2 \zeta_{4}$ |
$2$ | $4$ | $(1,5,8,2)(3,4,7,6)$ | $0$ |
$2$ | $8$ | $(1,7,2,6,8,3,5,4)$ | $0$ |
$2$ | $8$ | $(1,6,5,7,8,4,2,3)$ | $0$ |
$2$ | $8$ | $(1,6,2,3,8,4,5,7)$ | $0$ |
$2$ | $8$ | $(1,3,5,6,8,7,2,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.