Basic invariants
Dimension: | $2$ |
Group: | $C_8:C_2$ |
Conductor: | \(225\)\(\medspace = 3^{2} \cdot 5^{2} \) |
Artin stem field: | Galois closure of 8.4.56953125.1 |
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
$f(x)$ | $=$ |
\( x^{8} - x^{7} + 2x^{6} + 2x^{5} - 5x^{4} + 13x^{3} - 13x^{2} + x + 1 \)
|
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 14 + 36\cdot 61 + 7\cdot 61^{2} + 26\cdot 61^{3} + 33\cdot 61^{4} +O(61^{5})\)
|
$r_{ 2 }$ | $=$ |
\( 21 + 47\cdot 61 + 14\cdot 61^{2} + 3\cdot 61^{3} + 18\cdot 61^{4} +O(61^{5})\)
|
$r_{ 3 }$ | $=$ |
\( 24 + 3\cdot 61 + 60\cdot 61^{2} + 3\cdot 61^{3} + 43\cdot 61^{4} +O(61^{5})\)
|
$r_{ 4 }$ | $=$ |
\( 37 + 17\cdot 61 + 12\cdot 61^{2} + 59\cdot 61^{3} + 42\cdot 61^{4} +O(61^{5})\)
|
$r_{ 5 }$ | $=$ |
\( 44 + 33\cdot 61 + 37\cdot 61^{2} + 20\cdot 61^{3} + 26\cdot 61^{4} +O(61^{5})\)
|
$r_{ 6 }$ | $=$ |
\( 52 + 61 + 60\cdot 61^{2} + 24\cdot 61^{3} + 23\cdot 61^{4} +O(61^{5})\)
|
$r_{ 7 }$ | $=$ |
\( 56 + 28\cdot 61 + 52\cdot 61^{2} + 16\cdot 61^{3} + 38\cdot 61^{4} +O(61^{5})\)
|
$r_{ 8 }$ | $=$ |
\( 58 + 13\cdot 61 + 60\cdot 61^{2} + 27\cdot 61^{3} + 18\cdot 61^{4} +O(61^{5})\)
|
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 | Complex conjugation |
$1$ | $1$ | $()$ | $2$ | |
$1$ | $2$ | $(1,8)(2,4)(3,5)(6,7)$ | $-2$ | |
$2$ | $2$ | $(1,8)(3,5)$ | $0$ | ✓ |
$1$ | $4$ | $(1,5,8,3)(2,7,4,6)$ | $2 \zeta_{4}$ | |
$1$ | $4$ | $(1,3,8,5)(2,6,4,7)$ | $-2 \zeta_{4}$ | |
$2$ | $4$ | $(1,5,8,3)(2,6,4,7)$ | $0$ | |
$2$ | $8$ | $(1,4,5,6,8,2,3,7)$ | $0$ | |
$2$ | $8$ | $(1,6,3,4,8,7,5,2)$ | $0$ | |
$2$ | $8$ | $(1,2,3,6,8,4,5,7)$ | $0$ | |
$2$ | $8$ | $(1,6,5,2,8,7,3,4)$ | $0$ |