Basic invariants
Dimension: | $3$ |
Group: | $\GL(3,2)$ |
Conductor: | \(299475\)\(\medspace = 3^{2} \cdot 5^{2} \cdot 11^{3} \) |
Artin stem field: | Galois closure of 7.3.9965030625.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $\PSL(2,7)$ |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $\GL(3,2)$ |
Projective stem field: | Galois closure of 7.3.9965030625.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - x^{6} + 13x^{5} - 23x^{4} + 8x^{3} - 23x^{2} + 21x + 9 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \( x^{3} + 2x + 11 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 4 a^{2} + 12 a + 1 + \left(11 a^{2} + 7\right)\cdot 13 + \left(9 a^{2} + 2 a + 10\right)\cdot 13^{2} + \left(3 a^{2} + 11 a + 5\right)\cdot 13^{3} + \left(10 a + 11\right)\cdot 13^{4} + \left(a^{2} + 8 a + 6\right)\cdot 13^{5} + \left(3 a^{2} + 4 a + 8\right)\cdot 13^{6} + \left(7 a^{2} + 8 a + 1\right)\cdot 13^{7} + \left(11 a^{2} + 2 a + 6\right)\cdot 13^{8} +O(13^{9})\)
$r_{ 2 }$ |
$=$ |
\( 11 a + 11 + \left(2 a^{2} + 8 a + 8\right)\cdot 13 + \left(4 a^{2} + 3 a + 12\right)\cdot 13^{2} + \left(3 a^{2} + a + 5\right)\cdot 13^{3} + \left(4 a^{2} + 8 a + 4\right)\cdot 13^{4} + \left(2 a^{2} + 9\right)\cdot 13^{5} + \left(9 a^{2} + 8 a + 5\right)\cdot 13^{6} + \left(12 a^{2} + 6 a + 11\right)\cdot 13^{7} + \left(11 a^{2} + a + 8\right)\cdot 13^{8} +O(13^{9})\)
| $r_{ 3 }$ |
$=$ |
\( 10 a^{2} + 3 a + 7 + \left(3 a^{2} + 7 a + 2\right)\cdot 13 + \left(10 a^{2} + a + 12\right)\cdot 13^{2} + \left(10 a^{2} + 2\right)\cdot 13^{3} + \left(12 a^{2} + 10 a + 7\right)\cdot 13^{4} + \left(10 a^{2} + a + 3\right)\cdot 13^{5} + \left(9 a^{2} + 12 a + 2\right)\cdot 13^{6} + \left(5 a^{2} + 6 a + 2\right)\cdot 13^{7} + \left(3 a^{2} + 2 a + 6\right)\cdot 13^{8} +O(13^{9})\)
| $r_{ 4 }$ |
$=$ |
\( 2 a^{2} + 2 a + 7 + \left(5 a^{2} + 8 a + 7\right)\cdot 13 + \left(3 a^{2} + 5 a + 10\right)\cdot 13^{2} + \left(5 a^{2} + 5 a + 7\right)\cdot 13^{3} + \left(2 a^{2} + 10 a + 5\right)\cdot 13^{4} + \left(5 a^{2} + 11 a + 12\right)\cdot 13^{5} + \left(8 a^{2} + 6\right)\cdot 13^{6} + \left(11 a^{2} + 7\right)\cdot 13^{7} + \left(3 a^{2} + 4 a + 4\right)\cdot 13^{8} +O(13^{9})\)
| $r_{ 5 }$ |
$=$ |
\( 3 a^{2} + 12 a + 2 + \left(7 a^{2} + 9 a + 7\right)\cdot 13 + \left(11 a^{2} + 7 a + 9\right)\cdot 13^{2} + \left(11 a^{2} + 11 a + 8\right)\cdot 13^{3} + \left(8 a^{2} + 7 a + 10\right)\cdot 13^{4} + \left(12 a^{2} + 10 a + 5\right)\cdot 13^{5} + \left(6 a^{2} + 5 a + 11\right)\cdot 13^{6} + \left(7 a^{2} + 12 a + 8\right)\cdot 13^{7} + \left(10 a^{2} + 8 a + 2\right)\cdot 13^{8} +O(13^{9})\)
| $r_{ 6 }$ |
$=$ |
\( 7 a^{2} + 12 a + 5 + \left(9 a^{2} + 3 a\right)\cdot 13 + \left(12 a^{2} + 5 a + 10\right)\cdot 13^{2} + \left(3 a^{2} + 9 a + 1\right)\cdot 13^{3} + \left(10 a^{2} + 4 a + 3\right)\cdot 13^{4} + \left(6 a^{2} + 5 a + 10\right)\cdot 13^{5} + \left(a^{2} + 7 a + 10\right)\cdot 13^{6} + \left(7 a^{2} + 4 a + 5\right)\cdot 13^{7} + \left(10 a^{2} + 6 a\right)\cdot 13^{8} +O(13^{9})\)
| $r_{ 7 }$ |
$=$ |
\( 7 + 5\cdot 13 + 12\cdot 13^{2} + 5\cdot 13^{3} + 9\cdot 13^{4} + 3\cdot 13^{5} + 6\cdot 13^{6} + 13^{7} + 10\cdot 13^{8} +O(13^{9})\)
| |
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 |
$1$ | $1$ | $()$ | $3$ |
$21$ | $2$ | $(1,4)(3,7)$ | $-1$ |
$56$ | $3$ | $(1,5,4)(3,7,6)$ | $0$ |
$42$ | $4$ | $(1,2)(3,5,6,4)$ | $1$ |
$24$ | $7$ | $(1,3,7,5,6,4,2)$ | $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ |
$24$ | $7$ | $(1,5,2,7,4,3,6)$ | $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.