Basic invariants
| Dimension: | $3$ |
| Group: | $\GL(3,2)$ |
| Conductor: | \(17606416\)\(\medspace = 2^{4} \cdot 1049^{2} \) |
| Artin stem field: | Galois closure of 7.3.70425664.1 |
| 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 7.3.70425664.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 3x^{6} + x^{5} + 3x^{4} - 4x^{3} + 4x^{2} - 4 \)
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The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{3} + 4x + 17 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 12 a^{2} + 7 a + 9 + \left(4 a^{2} + 16 a + 7\right)\cdot 19 + \left(a + 13\right)\cdot 19^{2} + \left(5 a^{2} + 15 a + 7\right)\cdot 19^{3} + \left(6 a^{2} + 10 a + 17\right)\cdot 19^{4} + \left(3 a^{2} + 4 a + 17\right)\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 13 + 15\cdot 19^{2} + 5\cdot 19^{3} + 10\cdot 19^{4} + 18\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 16 a^{2} + 16 a + 18 + \left(7 a^{2} + 4 a + 12\right)\cdot 19 + \left(4 a^{2} + 17 a + 6\right)\cdot 19^{2} + \left(5 a^{2} + 11 a + 11\right)\cdot 19^{3} + \left(a^{2} + 3 a + 5\right)\cdot 19^{4} + \left(11 a^{2} + 11 a + 14\right)\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 4 a + 15 + \left(5 a^{2} + 9 a + 14\right)\cdot 19 + \left(16 a^{2} + 7 a + 11\right)\cdot 19^{2} + \left(9 a^{2} + 11 a + 1\right)\cdot 19^{3} + \left(4 a^{2} + 15 a\right)\cdot 19^{4} + \left(16 a^{2} + 8\right)\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 18 a^{2} + 6 a + 17 + \left(18 a^{2} + a + 10\right)\cdot 19 + \left(13 a^{2} + 9 a\right)\cdot 19^{2} + \left(7 a^{2} + 4 a + 18\right)\cdot 19^{3} + \left(a^{2} + 9 a + 5\right)\cdot 19^{4} + \left(a^{2} + 17 a\right)\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 7 a^{2} + 8 a + 2 + \left(9 a^{2} + 12 a + 1\right)\cdot 19 + \left(2 a^{2} + 9 a + 13\right)\cdot 19^{2} + \left(4 a^{2} + 11 a + 11\right)\cdot 19^{3} + \left(8 a^{2} + 11 a + 3\right)\cdot 19^{4} + \left(18 a^{2} + 13 a + 1\right)\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 4 a^{2} + 16 a + 5 + \left(11 a^{2} + 12 a + 9\right)\cdot 19 + \left(11 a + 15\right)\cdot 19^{2} + \left(6 a^{2} + 2 a\right)\cdot 19^{3} + \left(16 a^{2} + 6 a + 14\right)\cdot 19^{4} + \left(6 a^{2} + 9 a + 15\right)\cdot 19^{5} +O(19^{6})\)
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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 |
| $1$ | $1$ | $()$ | $3$ |
| $21$ | $2$ | $(1,5)(6,7)$ | $-1$ |
| $56$ | $3$ | $(1,4,5)(3,6,7)$ | $0$ |
| $42$ | $4$ | $(1,7,4,3)(2,6)$ | $1$ |
| $24$ | $7$ | $(1,6,2,7,4,3,5)$ | $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ |
| $24$ | $7$ | $(1,7,5,2,3,6,4)$ | $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.