Basic invariants
| Dimension: | $3$ |
| Group: | $\GL(3,2)$ |
| Conductor: | \(345744\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 7^{4} \) |
| Artin stem field: | Galois closure of 7.3.3320525376.3 |
| 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 7.3.3320525376.3 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 7x^{5} + 21x^{3} - 14x^{2} - 7x + 4 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 4 a^{2} + 13 a + 10 + \left(4 a^{2} + 18 a + 1\right)\cdot 19 + \left(8 a^{2} + 14 a + 17\right)\cdot 19^{2} + \left(14 a^{2} + 17 a + 9\right)\cdot 19^{3} + \left(7 a^{2} + 11 a + 18\right)\cdot 19^{4} + \left(9 a^{2} + 12 a + 8\right)\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 8 + 6\cdot 19 + 16\cdot 19^{2} + 9\cdot 19^{3} + 2\cdot 19^{4} + 6\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 17 a^{2} + 3 a + 18 + \left(9 a^{2} + a + 14\right)\cdot 19 + \left(a^{2} + 16 a + 9\right)\cdot 19^{2} + \left(7 a^{2} + 14 a + 12\right)\cdot 19^{3} + \left(14 a^{2} + 4 a + 1\right)\cdot 19^{4} + \left(17 a^{2} + 16 a + 17\right)\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 9 a^{2} + 15 a + 17 + \left(4 a^{2} + 13 a + 14\right)\cdot 19 + \left(5 a^{2} + 15 a + 2\right)\cdot 19^{2} + \left(8 a^{2} + 8 a + 6\right)\cdot 19^{3} + \left(4 a^{2} + 7 a + 3\right)\cdot 19^{4} + \left(13 a^{2} + 6 a\right)\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 6 a^{2} + 10 a + 9 + \left(10 a^{2} + 5 a + 11\right)\cdot 19 + \left(5 a^{2} + 7 a + 3\right)\cdot 19^{2} + \left(15 a^{2} + 11 a + 12\right)\cdot 19^{3} + \left(6 a^{2} + 18 a + 9\right)\cdot 19^{4} + \left(15 a^{2} + 18 a + 18\right)\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 16 a^{2} + 14 a + 9 + \left(12 a^{2} + 17 a + 16\right)\cdot 19 + \left(13 a^{2} + 13 a + 16\right)\cdot 19^{2} + \left(9 a^{2} + 9 a + 6\right)\cdot 19^{3} + \left(12 a^{2} + 6 a + 9\right)\cdot 19^{4} + \left(4 a + 9\right)\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 5 a^{2} + 2 a + 5 + \left(15 a^{2} + 10\right)\cdot 19 + \left(3 a^{2} + 8 a + 9\right)\cdot 19^{2} + \left(2 a^{2} + 13 a + 18\right)\cdot 19^{3} + \left(11 a^{2} + 7 a + 11\right)\cdot 19^{4} + \left(17 a + 15\right)\cdot 19^{5} +O(19^{6})\)
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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 |
| $1$ | $1$ | $()$ | $3$ |
| $21$ | $2$ | $(1,2)(5,6)$ | $-1$ |
| $56$ | $3$ | $(1,7,2)(3,6,5)$ | $0$ |
| $42$ | $4$ | $(2,3,7,6)(4,5)$ | $1$ |
| $24$ | $7$ | $(1,3,7,6,4,5,2)$ | $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ |
| $24$ | $7$ | $(1,6,2,7,5,3,4)$ | $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.