Basic invariants
Dimension: | $3$ |
Group: | $C_7:C_3$ |
Conductor: | \(97969\)\(\medspace = 313^{2} \) |
Artin stem field: | Galois closure of 7.7.9597924961.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_7:C_3$ |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $C_7:C_3$ |
Projective stem field: | Galois closure of 7.7.9597924961.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - x^{6} - 15x^{5} + 20x^{4} + 33x^{3} - 22x^{2} - 32x - 8 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \( x^{3} + 2x + 9 \)
Roots:
$r_{ 1 }$ | $=$ |
\( a + 8 + \left(8 a + 7\right)\cdot 11 + \left(a^{2} + 9\right)\cdot 11^{2} + 6 a^{2} 11^{3} + \left(6 a^{2} + 10 a + 8\right)\cdot 11^{4} + \left(10 a^{2} + 10 a + 7\right)\cdot 11^{5} + \left(5 a + 10\right)\cdot 11^{6} + \left(7 a^{2} + 7 a + 5\right)\cdot 11^{7} + \left(7 a^{2} + 2 a + 5\right)\cdot 11^{8} + \left(a^{2} + a + 4\right)\cdot 11^{9} +O(11^{10})\)
$r_{ 2 }$ |
$=$ |
\( a^{2} + a + 2 + \left(9 a + 4\right)\cdot 11 + \left(7 a^{2} + a + 10\right)\cdot 11^{2} + \left(3 a^{2} + 6 a + 4\right)\cdot 11^{3} + \left(8 a^{2} + a + 10\right)\cdot 11^{4} + \left(5 a^{2} + 3 a + 4\right)\cdot 11^{5} + \left(6 a^{2} + 3 a + 3\right)\cdot 11^{6} + \left(10 a^{2} + 10 a + 3\right)\cdot 11^{7} + \left(2 a^{2} + 7 a + 10\right)\cdot 11^{8} + \left(6 a^{2} + 6\right)\cdot 11^{9} +O(11^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 8 a^{2} + 10 a + 2 + \left(4 a^{2} + 10 a + 7\right)\cdot 11 + \left(2 a^{2} + 6 a + 3\right)\cdot 11^{2} + \left(4 a^{2} + 7 a\right)\cdot 11^{3} + \left(7 a^{2} + 5 a + 10\right)\cdot 11^{4} + \left(4 a^{2} + 6 a + 7\right)\cdot 11^{5} + \left(6 a^{2} + 7 a\right)\cdot 11^{6} + \left(6 a + 7\right)\cdot 11^{7} + \left(6 a^{2} + 3 a + 8\right)\cdot 11^{8} + \left(7 a^{2} + 2 a\right)\cdot 11^{9} +O(11^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 9 a^{2} + 4 a + 7 + \left(4 a^{2} + 3\right)\cdot 11 + \left(7 a^{2} + 5 a + 10\right)\cdot 11^{2} + \left(6 a^{2} + 2 a + 10\right)\cdot 11^{3} + \left(2 a^{2} + 10\right)\cdot 11^{4} + \left(3 a^{2} + 8 a + 5\right)\cdot 11^{5} + 3\cdot 11^{6} + \left(7 a^{2} + 4 a + 8\right)\cdot 11^{7} + \left(9 a^{2} + 3 a + 9\right)\cdot 11^{8} + \left(5 a^{2} + 5 a + 5\right)\cdot 11^{9} +O(11^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 5 a^{2} + 8 a + 9 + \left(a^{2} + 10 a + 2\right)\cdot 11 + \left(a^{2} + 9 a + 9\right)\cdot 11^{2} + 5\cdot 11^{3} + \left(a^{2} + 5 a + 1\right)\cdot 11^{4} + \left(3 a^{2} + 7 a + 2\right)\cdot 11^{5} + \left(4 a^{2} + 2 a + 5\right)\cdot 11^{6} + \left(3 a^{2} + 3\right)\cdot 11^{7} + \left(6 a^{2} + 4 a + 5\right)\cdot 11^{8} + \left(8 a^{2} + 3 a + 9\right)\cdot 11^{9} +O(11^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 3 + 7\cdot 11 + 6\cdot 11^{2} + 4\cdot 11^{3} + 11^{4} + 3\cdot 11^{5} + 6\cdot 11^{6} + 2\cdot 11^{7} + 11^{8} + 10\cdot 11^{9} +O(11^{10})\)
| $r_{ 7 }$ |
$=$ |
\( 10 a^{2} + 9 a + 3 + \left(10 a^{2} + 4 a\right)\cdot 11 + \left(2 a^{2} + 8 a + 5\right)\cdot 11^{2} + \left(a^{2} + 4 a + 5\right)\cdot 11^{3} + \left(7 a^{2} + 10 a + 1\right)\cdot 11^{4} + \left(5 a^{2} + 7 a + 1\right)\cdot 11^{5} + \left(3 a^{2} + a + 3\right)\cdot 11^{6} + \left(4 a^{2} + 4 a + 2\right)\cdot 11^{7} + 3\cdot 11^{8} + \left(3 a^{2} + 9 a + 6\right)\cdot 11^{9} +O(11^{10})\)
| |
Generators of the action on the roots $r_1, \ldots, r_{ 7 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 7 }$ | Character value |
$1$ | $1$ | $()$ | $3$ |
$7$ | $3$ | $(1,2,7)(3,5,4)$ | $0$ |
$7$ | $3$ | $(1,7,2)(3,4,5)$ | $0$ |
$3$ | $7$ | $(1,3,4,6,2,7,5)$ | $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ |
$3$ | $7$ | $(1,6,5,4,7,3,2)$ | $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.